tag:blogger.com,1999:blog-2212338509380989642024-03-18T21:16:34.636-07:00OpenAlgebra.com: Free Algebra Study Guide & Video TutorialsExplore our comprehensive free algebra study guide with step-by-step notes, video tutorials, and interactive content. Master linear equations, polynomials, and more.John Reddenhttp://www.blogger.com/profile/02231088988057526726noreply@blogger.comBlogger155125tag:blogger.com,1999:blog-221233850938098964.post-35905058104340844802023-07-11T12:12:00.003-07:002023-08-13T08:17:13.518-07:00Free Algebra Study Guide with VideosWelcome to OpenAlgebra.com, your comprehensive online destination for mastering algebra. Dive deep into our extensive collection of step-by-step tutorials, complemented by illustrative YouTube videos for a visual learning experience. Whether you're a student seeking supplemental material beyond your textbooks or a teacher looking for a rich resource of worked-out math problems, our platform offers a treasure trove of algebraic knowledge tailored to enhance your understanding.<br />
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<tr><td valign="top" width="50%"><b><span style="font-size: large;"><a href="https://www.openalgebra.com/2012/11/visual-real-numbers-and-their-operations.html">Chapter 1</a> - Real Numbers and Their Operations</span></b><br />
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1.1 <a href="https://www.openalgebra.com/2012/11/real-numbers-and-number-line.html">Real Numbers and The Number Line</a><br />
1.2 <a href="https://www.openalgebra.com/2012/11/adding-and-subtracting-integers.html">Adding and Subtracting Integers</a><br />
1.3 <a href="https://www.openalgebra.com/2012/11/multiplying-and-dividing-integers.html">Multiplying and Dividing Integers</a><br />
1.4 <a href="https://www.openalgebra.com/2012/11/fractions.html">Fractions</a><br />
1.5 <a href="https://www.openalgebra.com/2012/11/review-of-decimals-and-percents.html">Review of Decimals and Percents</a><br />
1.6 <a href="https://www.openalgebra.com/2012/11/exponents-and-square-roots.html">Exponents and Square Roots</a><br />
1.7 <a href="https://www.openalgebra.com/2012/11/order-of-operations.html">Order of Operations</a><br />
1.8 <a href="https://www.openalgebra.com/2012/11/chapter-1-sample-test-questions.html">Sample Exam Questions</a><br />
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<b><span style="font-size: large;"><a href="https://www.openalgebra.com/2013/02/visual-linear-equations-and-inequalities.html">Chapter 2</a> - Linear Equations and Inequalities</span></b><br />
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2.1 <a href="https://www.openalgebra.com/p/introduction-to-algebra.html">Introduction to Algebra</a><br />
2.2 <a href="https://www.openalgebra.com/2012/11/introduction-to-algebra.html" target="_blank">Simplifying Algebraic Expressions</a><br />
2.3 <a href="https://www.openalgebra.com/2012/11/solving-linear-equations-part-i.html" target="_blank">Linear Equations: Part I</a><br />
2.4 <a href="https://www.openalgebra.com/2012/11/solving-linear-equations-part-ii.html" target="_blank">Linear Equations: Part II</a><br />
2.5 <a href="https://www.openalgebra.com/2012/11/applications-of-linear-equations.html" target="_blank">Applications of Linear Equations</a><br />
2.6 <a href="https://www.openalgebra.com/2012/11/ratio-and-proportion-applications.html" target="_blank">Ratio and Proportion Applications</a><br />
2.7 <a href="https://www.openalgebra.com/2012/11/introduction-to-inequalities-and.html" target="_blank">Introduction to Inequalities and Interval Notation</a><br />
2.8 <a href="https://www.openalgebra.com/2012/11/linear-inequalities-one-variable.html" target="_blank">Linear Inequalities (one variable)</a><br />
2.9 <a href="https://www.openalgebra.com/2012/11/chapter-2-sample-test-questions.html" target="">Review Exercises and Sample Exam</a><br />
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<b><span style="font-size: large;"><a href="https://www.openalgebra.com/2013/02/visual-graphing-lines.html">Chapter 3</a> - Graphing Lines</span></b><br />
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3.1 <a href="https://www.openalgebra.com/2012/11/rectangular-coordinate-system.html">Rectangular Coordinate System</a><br />
3.2 <a href="https://www.openalgebra.com/2012/11/graph-by-plotting-points.html">Graph by Plotting Points</a><br />
3.3 <a href="https://www.openalgebra.com/2012/11/graph-using-intercepts.html">Graph using Intercepts</a><br />
3.4 <a href="https://www.openalgebra.com/2012/11/graph-using-y-intercept-and-slope.html">Graph using the y-intercept and Slope</a><br />
3.5 <a href="https://www.openalgebra.com/2012/11/finding-linear-equations.html">Finding Linear Equations</a><br />
3.6 <a href="https://www.openalgebra.com/2012/11/parallel-and-perpendicular-lines.html">Parallel and Perpendicular Lines</a><br />
3.7 <a href="https://www.openalgebra.com/2012/11/introduction-to-functions.html">Introduction to Functions</a><br />
3.8 <a href="https://www.openalgebra.com/2012/11/linear-inequalities-two-variables.html">Linear Inequalities (Two Variables)</a><br />
3.9 <a href="https://www.openalgebra.com/2012/11/chapter-3-sample-test-questions.html">Review Exercises and Sample Exam</a><br />
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<b><span style="font-size: large;"><a href="https://www.openalgebra.com/2013/02/visual-solving-linear-equations.html">Chapter 4</a> - Solving Linear Systems</span></b><br />
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4.1 <a href="https://www.openalgebra.com/2012/11/solving-linear-systems-by-graphing.html">Solving Linear Systems by Graphing</a><br />
4.2 <a href="https://www.openalgebra.com/2012/11/solving-linear-systems-by-substitution.html">Solving Linear Systems by Substitution</a><br />
4.3 <a href="https://www.openalgebra.com/2012/11/solving-linear-systems-by-elimination.html">Solving Linear Systems by Elimination</a><br />
4.4 <a href="https://www.openalgebra.com/2012/11/applications-of-linear-systems.html">Applications of Linear Systems</a><br />
4.5 <a href="https://www.openalgebra.com/2012/11/systems-of-linear-inequalities-two.html">Solving Systems of Linear Inequalities (Two Variables)</a><br />
4.6 <a href="https://www.openalgebra.com/2012/11/chapter-4-sample-test-questions.html">Review Exercises and Sample Exam</a><br />
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[ <a href="https://www.openalgebra.com/2013/02/elementary-algebra-exam-1.html">Elementary Algebra Exam #1 Solutions</a> ]<br />
[ <a href="https://www.openalgebra.com/2013/03/elementary-algebra-exam-2.html">Elementary Algebra Exam #2 Solutions</a> ]<br />
[ <a href="https://www.openalgebra.com/2013/04/elementary-algebra-exam-3.html">Elementary Algebra Exam #3 Solutions</a> ]<br />
[ <a href="https://www.openalgebra.com/2013/05/elementary-algebra-exam-4.html">Elementary Algebra Exam #4 Solutions</a> ]<br />
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<b><span style="font-size: large;"><a href="https://www.openalgebra.com/2013/02/visual-polynomials-and-their-operations.html">Chapter 5</a> - Polynomials and Their Operations</span></b><br />
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5.1 <a href="https://www.openalgebra.com/2012/11/rules-of-exponents.html">Rules of Exponents (Integer Exponents)</a><br />
5.2 <a href="https://www.openalgebra.com/2012/11/introduction-to-polynomials.html">Introduction to Polynomials and Evaluating</a><br />
5.3 <a href="https://www.openalgebra.com/2012/11/adding-and-subtracting-polynomials.html">Adding and Subtracting Polynomials</a><br />
5.4 <a href="https://www.openalgebra.com/2012/11/multiplying-polynomials.html">Multiplying Polynomials and Special Products</a><br />
5.5 <a href="https://www.openalgebra.com/2012/11/dividing-polynomials.html">Dividing Polynomials</a> (<a href="https://www.openalgebra.com/2013/10/synthetic-division.html" target="_blank">Synthetic Division</a>)<br />
5.6 <a href="https://www.openalgebra.com/2012/11/negative-exponents.html" target="_blank">Negative Exponents and Scientific Notation</a><br />
5.7 <a href="https://www.openalgebra.com/2012/11/chapter-5-sample-test-questions.html">Review Exercises and Sample Exam</a><br />
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<b><span style="font-size: large;"><a href="https://www.openalgebra.com/2013/02/visual-factoring-and-solving-by.html">Chapter 6</a> - Factoring and Solving by Factoring</span></b><br />
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6.1 <a href="https://www.openalgebra.com/2012/11/introduction-to-factoring-and-factor-by.html">Introduction to Factoring</a><br />
6.2 <a href="https://www.openalgebra.com/2012/11/factoring-trinomials-of-form-x2-bx-c.html">Factoring Trinomials</a><br />
6.3 <a href="https://www.openalgebra.com/2012/11/factoring-trinomials-of-form-ax2-bx-c.html">Factoring Trinomials ax^2 + bx + c</a><br />
6.4 <a href="https://www.openalgebra.com/2012/11/factoring-special-binomials.html">Factoring Binomials</a><br />
6.5 <a href="https://www.openalgebra.com/2012/11/general-guidelines-for-factoring.html">General Guidelines for Factoring Polynomials</a><br />
6.6 <a href="https://www.openalgebra.com/2012/11/solving-equations-by-factoring.html">Solving Equations by Factoring</a><br />
6.7 <a href="https://www.openalgebra.com/2012/11/applications-involving-quadratic.html">Applications involving Quadratic Equations</a><br />
6.8 <a href="https://www.openalgebra.com/2012/11/chapter-6-sample-test-questions.html">Review Exercises and Sample Exam</a><br />
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<b><span style="font-size: large;"><a href="https://www.openalgebra.com/2013/02/visual-rational-expressions-and.html">Chapter 7</a> - Rational Expressions and Equations</span></b><br />
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7.1 <a href="https://www.openalgebra.com/2012/11/simplifying-rational-expressions.html">Simplifying Rational Expressions</a><br />
7.2 <a href="https://www.openalgebra.com/2012/11/multiplying-and-dividing-rational.html">Multiplying and Dividing Rational Expressions</a><br />
7.3 <a href="https://www.openalgebra.com/2012/11/adding-and-subtracting-rational.html">Adding and Subtracting Rational Expressions</a><br />
7.4 <a href="https://www.openalgebra.com/2012/11/complex-rational-expressions.html">Complex Fractions</a><br />
7.5 <a href="https://www.openalgebra.com/2012/11/solving-rational-equations.html">Solving Rational Equations</a><br />
7.6 <a href="https://www.openalgebra.com/2012/11/applications-of-rational-equations.html">Applications of Rational Equations</a><br />
7.7 <a href="https://www.openalgebra.com/2012/11/variation.html">Variation</a><br />
7.8 <a href="https://www.openalgebra.com/2012/11/chapter-7-sample-test-questions.html">Review Exercises and Sample Exam</a><br />
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Embark on an advanced journey with our <b>Intermediate Algebra</b> (IA) section, also known as <b>Algebra 2</b>. Delve into intricate topics, unravel complex equations, and enhance your mathematical prowess. Our meticulously crafted resources, combined with illustrative video tutorials, ensure a deeper understanding of algebraic concepts, bridging the gap between foundational knowledge and advanced applications. Whether you're gearing up for a challenging exam or simply eager to expand your algebraic horizons, our Intermediate Algebra section is tailored to meet your needs.<br />
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<tr><td valign="top"><b><span style="font-size: large;"><a href="https://www.openalgebra.com/2013/03/visual-radical-expressions-and-equations.html">IA Chapter</a> - Radical Expressions and Equations</span></b><br />
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1 <a href="https://www.openalgebra.com/2012/11/radicals.html">Radicals</a><br />
2 <a href="https://www.openalgebra.com/2012/11/simplifying-radical-expressions.html">Simplifying Radical Expressions</a><br />
3 <a href="https://www.openalgebra.com/2012/11/adding-and-subtracting-radical.html">Adding and Subtracting Radical Expressions</a><br />
4 <a href="https://www.openalgebra.com/2012/11/multiplying-and-dividing-radical.html">Multiplying and Dividing Radical Expressions</a><br />
5 <a href="https://www.openalgebra.com/2012/11/rational-exponents.html">Rational Exponents</a><br />
6 <a href="https://www.openalgebra.com/2012/11/solving-radical-equations.html">Solving Radical Equations</a><br />
7 <a href="https://www.openalgebra.com/2013/03/complex-numbers.html">Complex Numbers</a><br />
8 <a href="https://www.openalgebra.com/2012/11/chapter-8-sample-test-questions.html">Review Exercises and Sample Exam</a><br />
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<b><span style="font-size: large;">IA Chapter - Solving Quadratic Equations and Graphing Parabolas</span></b><br />
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1 <a href="https://www.openalgebra.com/2012/11/solving-quadratic-equations-and.html">Extracting Square Roots</a><br />
2 <a href="https://www.openalgebra.com/2012/11/completing-square.html">Completing the Square</a><br />
3 <a href="https://www.openalgebra.com/2012/11/quadratic-formula.html">Quadratic Formula</a><br />
4 <a href="https://www.openalgebra.com/2012/11/guidelines-for-solving-quadratic.html">Guidelines for Solving Quadratic Equations and Applications</a><br />
5 <a href="https://www.openalgebra.com/2013/10/solving-equations-quadratic-in-form.html">Solving Equations Quadratic in Form</a><br />
6 <a href="https://www.openalgebra.com/2012/11/graphing-parabolas.html">Graphing Parabolas</a><br />
7 <a href="https://www.openalgebra.com/2012/11/complex-numbers-and-complex-solutions.html">Complex Numbers and Complex Solutions</a><br />
8 <a href="https://www.openalgebra.com/2012/11/chapter-9-sample-test-questions.html">Review Exercises and Sample Exam</a><br />
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<b><span style="font-size: large;">IA Chapter - <a href="https://www.openalgebra.com/2019/07/solving-absolute-value-equations.html">Solving Inequalities</a> </span></b><br />
1 <a href="https://www.openalgebra.com/2012/11/absolute-value-equations.html">Absolute Value Equations</a><br />
2 <a href="https://www.openalgebra.com/2012/11/absolute-value-inequalities.html">Absolute Value Inequalities</a><br />
3 <a href="https://www.openalgebra.com/2012/11/quadratic-inequalities.html">Quadratic Inequalities</a><br />
4 <a href="https://www.openalgebra.com/2012/11/polynomial-and-rational-inequalities.html">Polynomial and Rational Inequalities</a><br />
<br />
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[ <a href="https://www.openalgebra.com/2013/02/intermediate-algebra-exam-1.html" target="_blank">Intermediate Algebra Exam #1 Solutions</a> ]<br />
[ <a href="https://www.openalgebra.com/2013/03/intermediate-algebra-exam-2.html" target="">Intermediate Algebra Exam #2 Solutions</a> ]<br />
[ <a href="https://www.openalgebra.com/2013/04/intermediate-algebra-exam-3.html">Intermediate Algebra Exam #3 Solutions</a> ]<br />
[ <a href="https://www.openalgebra.com/2013/05/intermediate-algebra-exam-4.html">Intermediate Algebra Exam #4 Solutions</a> ]<br />
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--- <a href="https://docs.google.com/file/d/0ByXp6Ibu-fCyS0k1SmVIWWMwRG8/edit?usp=sharing" target="_blank">IA Sample Final Exam Ch. 1- 7</a> ---</td>
<td valign="top"><b><span style="font-size: large;">IA Chapter - <a href="https://www.openalgebra.com/2019/08/graphing-basic-functions-with.html">Graphing Basic Functions</a></span></b>
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1 <a href="https://www.openalgebra.com/2013/07/relations-graphs-and-functions.html">Relations, Graphs, and Functions</a><br />
2 <a href="https://www.openalgebra.com/2013/02/graphing-basic-functions.html">Graphing the Basic Functions</a><br />
3 <a href="https://www.openalgebra.com/2013/02/graphing-functions-using-transformations.html">Using Transformations to Graph Functions</a><br />
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<b><span style="font-size: large;">IA Chapter - <a href="https://www.openalgebra.com/2013/10/logarithms-exponentials-and-equations.html">Exponential Equations and Logarithms</a></span></b><br />
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1 <a href="https://www.openalgebra.com/2012/11/function-composition.html">Function Composition</a><br />
2 <a href="https://www.openalgebra.com/2013/01/inverse-functions.html">Inverse Functions</a><br />
3 <a href="https://www.openalgebra.com/2012/11/exponential-functions-and-their-graphs.html">Exponential Functions and Their Graphs</a><br />
4 <a href="https://www.openalgebra.com/2012/11/the-natural-exponential-function.html">The Natural Exponential Function</a><br />
5 <a href="https://www.openalgebra.com/2012/11/logarithmic-functions.html">Logarithmic Functions</a><br />
6 <a href="https://www.openalgebra.com/2013/07/graphing-logarithmic-functions.html">Graphing Logarithmic Functions</a><br />
7 <a href="https://www.openalgebra.com/2012/11/change-of-base-formula.html">Change of Base Formula</a><br />
8 <a href="https://www.openalgebra.com/2012/11/properties-of-logarithm.html">Properties of the Logarithm</a><br />
9 <a href="https://www.openalgebra.com/2012/11/solving-exponential-equations.html">Solving Exponential Equations</a><br />
10 <a href="https://www.openalgebra.com/2012/11/solving-logarithmic-equations.html">Solving Logarithmic Equations</a><br />
11 Applications<br />
<a href="https://www.openalgebra.com/2013/07/interest-problems.html">Compound Interest Problems</a><br />
<a href="https://www.openalgebra.com/2013/07/exponential-growth-and-decay.html">Exponential Growth/Decay</a><br />
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<b><span style="font-size: large;">IA Chapter - <a href="https://www.openalgebra.com/2019/07/graphing-conic-section-in-standard-and.html">Graphing Conic Sections</a></span></b><br />
<br />
1. <a href="https://www.openalgebra.com/2013/06/distance-and-midpoint-formulas.html">Distance and Midpoint Formulas</a><br />
2. <a href="https://www.openalgebra.com/2013/04/circles.html">Circles</a><br />
3. <a href="https://www.openalgebra.com/2013/04/ellipses.html">Ellipses</a><br />
4. <a href="https://www.openalgebra.com/2013/05/hyperbolas.html">Hyperbolas</a><br />
5. <a href="https://www.openalgebra.com/2013/07/solving-nonlinear-systems.html">Solving Nonlinear Systems</a><br />
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<h2 style="text-align: left;">Factoring Trinomials of the Form \(x^2 - bx - c\)</h2>
<p>Factoring trinomials, which are polynomials with three terms, can initially seem challenging. However, with practice, it becomes routine. If a trinomial factors, it will factor into the product of two binomials.</p>
<h3 style="text-align: left;">Steps to Factor Trinomials:</h3>
<ol>
<li><b>Factor the first term:</b> \(x^2 = x \cdot x\).</li>
<li><b>Factor the last term:</b> Choose factors that multiply to obtain the last term and subtract to obtain the middle term.</li>
<li><b>Determine the signs:</b> by adding or subtracting the product of the inner and outer terms.</li>
<li><b>Check by multiplying.</b></li>
</ol>
<h3 style="text-align: left;">Example:</h3>
<p>Factor the trinomial \(x^2 - 3x - 10\).</p>
<ol>
<li>Factor the first term: \(x^2 = x \cdot x\).</li>
<li>Factor the last term: \(-10 = -5 \cdot 2\) and \(-5 + 2 = -3\), which is our middle term.</li>
<li>Determine the signs: One sign is positive and the other is negative because \(-5 + 2 = -3\).</li>
<li>Check by multiplying: \((x - 5)(x + 2) = x^2 - 3x - 10\).</li>
</ol>
<p>So, \(x^2 - 3x - 10 = (x - 5)(x + 2)\).</p>
<h3 style="text-align: left;">Exercises:</h3>
<p>Factor the following trinomials:</p>
<ol>
<li>\(x^2 - 5x - 6\)</li>
<li>\(x^2 - 7x + 10\)</li>
<li>\(x^2 - 6x - 16\)</li>
<li>\(x^2 - 8x - 9\)</li>
<li>\(x^2 - 3x - 10\)</li>
<li>\(x^2 - 4x - 21\)</li>
<li>\(x^2 - 5x - 14\)</li>
<li>\(x^2 - 6x - 16\)</li>
<li>\(x^2 - 7x - 18\)</li>
<li>\(x^2 - 9x - 22\)</li>
</ol>
<h3 style="text-align: left;">Solutions:</h3>
<ol>
<li>\(x^2 - 5x - 6 = (x - 6)(x + 1)\)</li>
<li>\(x^2 - 7x + 10 = (x - 2)(x - 5)\)</li>
<li>\(x^2 - 6x - 16 = (x - 8)(x + 2)\)</li>
<li>\(x^2 - 8x - 9 = (x - 9)(x + 1)\)</li>
<li>\(x^2 - 3x - 10 = (x - 5)(x + 2)\)</li>
<li>\(x^2 - 4x - 21 = (x - 7)(x + 3)\)</li>
<li>\(x^2 - 5x - 14 = (x - 7)(x + 2)\)</li>
<li>\(x^2 - 6x - 16 = (x - 8)(x + 2)\)</li>
<li>\(x^2 - 7x - 18 = (x - 9)(x + 2)\)</li>
<li>\(x^2 - 9x - 22 = (x - 11)(x + 2)\)</li>
</ol>
<p>Remember to always check your results by multiplying the factors. If the product matches the original trinomial, then the factoring is correct.</p>
John Reddenhttp://www.blogger.com/profile/02231088988057526726noreply@blogger.comtag:blogger.com,1999:blog-221233850938098964.post-17292304481761269702023-07-10T11:58:00.002-07:002023-07-11T12:02:08.259-07:00Factoring Lesson B by ChatGPT-4<script type="text/javascript" id="MathJax-script" async
src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js">
</script>
<h2 style="text-align: left;">Factoring Trinomials of the Form \(x^2 - bx + c\)</h2>
<p>Factoring trinomials, which are polynomials with three terms, can initially seem challenging. However, with practice, it becomes routine. If a trinomial factors, it will factor into the product of two binomials.</p>
<h3 style="text-align: left;">Steps to Factor Trinomials:</h3>
<ol>
<li><b>Factor the first term:</b> \(x^2 = x \cdot x\).</li>
<li><b>Factor the last term:</b> Choose factors that multiply to obtain the last term and add to obtain the middle term.</li>
<li><b>Determine the signs:</b> by adding or subtracting the product of the inner and outer terms.</li>
<li><b>Check by multiplying.</b></li>
</ol>
<h3 style="text-align: left;">Example:</h3>
<p>Factor the trinomial \(x^2 - 7x + 10\).</p>
<ol>
<li>Factor the first term: \(x^2 = x \cdot x\).</li>
<li>Factor the last term: \(10 = 5 \cdot 2\) and \(5 + 2 = 7\), which is our middle term.</li>
<li>Determine the signs: Both signs are negative because \(-5 - 2 = -7\).</li>
<li>Check by multiplying: \((x - 5)(x - 2) = x^2 - 7x + 10\).</li>
</ol>
<p>So, \(x^2 - 7x + 10 = (x - 5)(x - 2)\).</p>
<h3 style="text-align: left;">Exercises:</h3>
<p>Factor the following trinomials:</p>
<ol>
<li>\(x^2 - 5x + 6\)</li>
<li>\(x^2 - 9x + 20\)</li>
<li>\(x^2 - 7x + 10\)</li>
<li>\(x^2 - 11x + 30\)</li>
<li>\(x^2 - 8x + 15\)</li>
<li>\(x^2 - 10x + 21\)</li>
<li>\(x^2 - 6x + 8\)</li>
<li>\(x^2 - 12x + 35\)</li>
<li>\(x^2 - 15x + 56\)</li>
<li>\(x^2 - 13x + 40\)</li>
</ol>
<h3 style="text-align: left;">Solutions:</h3>
<ol>
<li>\(x^2 - 5x + 6 = (x - 3)(x - 2)\)</li>
<li>\(x^2 - 9x + 20 = (x - 5)(x - 4)\)</li>
<li>\(x^2 - 7x + 10 = (x - 5)(x - 2)\)</li>
<li>\(x^2 - 11x + 30 = (x - 6)(x - 5)\)</li>
<li>\(x^2 - 8x + 15 = (x - 5)(x - 3)\)</li>
<li>\(x^2 - 10x + 21 = (x - 7)(x - 3)\)</li>
<li>\(x^2 - 6x + 8 = (x - 4)(x - 2)\)</li>
<li>\(x^2 - 12x + 35 = (x - 7)(x - 5)\)</li>
<li>\(x^2 - 15x + 56 = (x - 8)(x - 7)\)</li>
<li>\(x^2 - 13x + 40 = (x - 8)(x - 5)\)</li>
</ol>
<p>Remember to always check your results by multiplying the factors. If the product matches the original trinomial, then the factoring is correct.</p>
John Reddenhttp://www.blogger.com/profile/02231088988057526726noreply@blogger.comtag:blogger.com,1999:blog-221233850938098964.post-74373386794952823822023-07-10T11:58:00.000-07:002023-07-11T11:58:31.913-07:00Factoring Lesson A by ChatGPT-4
<script async="" id="MathJax-script" src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js" type="text/javascript">
</script>
<h2 style="text-align: left;">Factoring Trinomials of the Form \(x^2 + bx + c\)</h2>
<p>Factoring trinomials, which are polynomials with three terms, can initially seem challenging. However, with practice, it becomes routine. If a trinomial factors, it will factor into the product of two binomials.</p>
<h3 style="text-align: left;">Steps to Factor Trinomials:</h3>
<ol>
<li><b>Factor the first term:</b> \(x^2 = x \cdot x\).</li>
<li><b>Factor the last term:</b> Choose factors that add or subtract to obtain the middle term.</li>
<li><b>Determine the signs:</b> by adding or subtracting the product of the inner and outer terms.</li>
<li><b>Check by multiplying.</b></li>
</ol>
<h3 style="text-align: left;">Example:</h3>
<p>Factor the trinomial \(x^2 + 7x + 12\).</p>
<ol>
<li>Factor the first term: \(x^2 = x \cdot x\).</li>
<li>Factor the last term: \(12 = 3 \cdot 4\) and \(3 + 4 = 7\), which is our middle term.</li>
<li>Determine the signs: Both signs are positive because \(+3 + 4 = +7\).</li>
<li>Check by multiplying: \((x + 3)(x + 4) = x^2 + 7x + 12\).</li>
</ol>
<p>So, \(x^2 + 7x + 12 = (x + 3)(x + 4)\).</p>
<h3 style="text-align: left;">Exercises:</h3>
<p>Factor the following trinomials:</p>
<ol>
<li>\(x^2 + 5x + 6\)</li>
<li>\(x^2 + 9x + 20\)</li>
<li>\(x^2 + 7x + 10\)</li>
<li>\(x^2 + 11x + 30\)</li>
<li>\(x^2 + 8x + 15\)</li>
<li>\(x^2 + 10x + 21\)</li>
<li>\(x^2 + 6x + 8\)</li>
<li>\(x^2 + 12x + 35\)</li>
<li>\(x^2 + 15x + 56\)</li>
<li>\(x^2 + 13x + 40\)</li>
</ol>
<h3 style="text-align: left;">Solutions:</h3>
<ol>
<li>\(x^2 + 5x + 6 = (x + 3)(x + 2)\)</li>
<li>\(x^2 + 9x + 20 = (x + 5)(x + 4)\)</li>
<li>\(x^2 + 7x + 10 = (x + 5)(x + 2)\)</li>
<li>\(x^2 + 11x + 30 = (x + 6)(x + 5)\)</li>
<li>\(x^2 + 8x + 15 = (x + 5)(x + 3)\)</li>
<li>\(x^2 + 10x + 21 = (x + 7)(x + 3)\)</li>
<li>\(x^2 + 6x + 8 = (x + 4)(x + 2)\)</li>
<li>\(x^2 + 12x + 35 = (x + 7)(x + 5)\)</li>
<li>\(x^2 + 15x + 56 = (x + 8)(x + 7)\)</li>
<li>\(x^2 + 13x + 40 = (x + 8)(x + 5)\)</li>
</ol>
<p>Remember to always check your results by multiplying the factors. If the product matches the original trinomial, then the factoring is correct.</p>
John Reddenhttp://www.blogger.com/profile/02231088988057526726noreply@blogger.comtag:blogger.com,1999:blog-221233850938098964.post-76419017403982779922022-04-01T12:58:00.001-07:002023-08-13T09:19:00.322-07:00Comprehensive Algebra Video Guide: Step-by-Step Solutions & Tutorials<div>
<div class="separator" style="clear: both; text-align: center;">
<img border="0" data-original-height="47" data-original-width="120" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgHCYcQnEnYVubfvdczeWXOMY53xCqMI5jFk75TTv1BD-TdGQK_wIF5XBke9NmJvf__Cm12D4FXPojzh4NYbmLIU7rfrLQNLvqgIfOq468_sj1b7Tq81Kp2bt6BQvkQxUakc-xWVRWdN0I/s1600/openAlgebra.png" /></div>
<div style="clear: both; text-align: center;">
<b><span style="color: #990000; font-size: large;">Click on a problem to see it worked out in YouTube.</span></b></div><div style="clear: both; text-align: center;"><span style="text-align: left;"><br /></span></div>Navigate the complexities of algebra with our extensive Algebra Video Compendium. From foundational concepts to advanced topics, our collection offers step-by-step video solutions on YouTube. Tailored to support the 'Intermediate Algebra' textbook by Flat World Knowledge, this visual guide simplifies algebraic learning, making it accessible and engaging for all.</div>
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1.1 <a href="https://www.openalgebra.com/2012/11/real-numbers-and-number-line.html">Real Numbers and The Number Line</a> <br />
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<a href="https://youtu.be/NNsdEV0jJqM"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh95r5CedHy447oQ8L5-gwPf-FTkv0FKgPEfweCN3A6uOKpfT0-6FaPBnGFCXtF-pvj2lKapiek0knZxCnn7Uo_1_j7XEOICYe7S7dMF4ggfcS7c0CvLdNQGZcAegORGm8xsFme_M_77DY/s1600/vid-0101.png" /></a><a href="https://youtu.be/4UaR4f4wNuE"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhuv52LPsAmnfy2xCv3lX4EIKdxo_KzGsFerHmJ5kG7Y2tz5bBMDCddtE7vClt1-tG8xbNBRW6-L8ZTH8tN45CsnuBHAn-iW4RpmS7OOlmqOsMiXgsxV_EnPvPtGFbBF-Rd0s1W5RV8k5E/s1600/vid-0102.png" /></a></div>
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1.2 <a href="https://www.openalgebra.com/2012/11/adding-and-subtracting-integers.html">Adding and Subtracting Integers</a> <br />
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<a href="https://youtu.be/l2y4rLS8I8s"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg7xwgnazCRi7aZz93aG1yy24zYe80N4Cmbg4DPNyoGaymjys2zvK1Escot53eSK0WoOlRWp1R9LEC21-001oX8sUsV6ULNp8Mnf3l1Ne8kvNqLmhXDr31H5GTQOS6WbYYqvTs9Jg5zHAA/s1600/vid-0103.png" /></a><a href="https://youtu.be/fGCRnW_i6I8"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhoXJ-wFPW4KAdTROJx9_lZMxQOjNtW8PjEWVOPU8VynE__4TwM-uVxvekf2Tq-E81plgXPEHsBWZ35_32G6JmfvDkGWnQCeopGg4nxp1IEr9X2HzmM6-3nwBF0gyhcEytF3pziDY-4_VE/s1600/vid-0104.png" /></a></div>
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1.3 <a href="https://www.openalgebra.com/2012/11/multiplying-and-dividing-integers.html">Multiplying and Dividing Integers</a><br />
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<a href="https://youtu.be/ftzGHykWq_A"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhTKcJxZHfg2woOLx3SppgiDMegpU0-GuzH6DRjhPoOZQge9QOe5WT-1JQzWGxJcnI6gVqFF9IfQ4jCDZpPNSXlOyG0sstFX0kWL8ApoDpfFKejqbgEMwm9kORLNo1L3JtvpjiCrrq6oxU/s1600/vid-0105.png" /></a><a href="https://youtu.be/mk2HoiqanmA"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrjipxlx2P1TopcKp_RQTSnm5Nf0l0iftWkMESO3gmoGkFbC8URw-6IBfLMiSpFmY3ORbPzVq36lIaSMiMRvqDtptfvZr1UxEBW9scGvQAAhz2wU6htjt9t_JvzPhGfTrbok3MSy9gCBA/s1600/vid-0106.png" /></a></div>
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1.4 <a href="https://www.openalgebra.com/2012/11/fractions.html">Fractions</a><br />
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<a href="https://youtu.be/D1JgMzcwC74"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhgX9BDwVd4VzH2LofmVDYQ4THFW4ZPJ8klLlgxarQgxGFSeXqNJen7I0YG6TZ1soSN4Q0u8GLscmY1OkUim_hxVj07gelCQfFNffqdrlUzn0DoL3x7kFLdzV2BPXBxzQkO_CuGCaEvmPE/s1600/vid-0107.png" /></a><a href="https://youtu.be/0nLzm9AWBDA"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgWvvcovrebKWU1XxHWNEUC6JGJeOxI-ESEm6xjl6cu49mzOnwMSvU_qH1kVpldUWM_vrG4khlMK0iaYfQsh5DvG-xVSAw6uTWapztFHCjpMmx-32cKpJsJ7lMXsrYCPyMk1JUnHfT0bqI/s1600/vid-0108.png" /></a><a href="https://youtu.be/T47YNYsxfp4"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg1e2m4z-wIkKxVzpOn-mz5LrQpFIcfzdibbEE-R_wKjvMqgQIB6HP3KAvA-8yAT0QdO_pAbAkWchdtLyAuv7alZvw7njx-_GgixpPnBJqClG2ZWV5lcdC0bY4np2VXNS-c1iL7RWKB8p0/s1600/vid-0109.png" /></a><a href="https://youtu.be/ojuJs-2YAMw"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiHCb9FCN40GYwoAFHpxqYP6JdkARz7UrhP8PeLMyspq7hCXWTgMhaup8J3CyCkDHo_cinEp_Tp-LR1h37xB3iIgiOmGwnw4Ima30MM3bM94E06ZUxCkQdoWGBFTDjdFjUPMbLK71WnzO8/s1600/vid-0110.png" /></a></div>
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1.5 <a href="https://www.openalgebra.com/2012/11/review-of-decimals-and-percents.html">Review of Decimals and Percents</a><br />
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<a href="https://youtu.be/NG8DrPm5qOc"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6xafV7303_Kba1TVk92mFOwyju3qVnrETUWFVUmmjDN8UTfVLoSrrIafoXuLE5nJYsaNKicnncPt86rbwh96a9VaSvpXb4yFkJsm3bqpyc0kA6NReKBzvSA2tUrfsJTjqZIsA3MxbUp8/s1600/vid-0111.png" /></a><a href="https://youtu.be/8Tbwx7ZdY1U"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWL4Thahu1wX0gtyT4ByTkfNBZrMCNU8Kz0R6cijx_XKrPwX1NOuvesWro1VboYJugaztoDCfupZR6MDD5qmb4pQ-226iTg3bGCIX87heS0LLpkiAbiwQXKQ8hHwnQa9qs9bD60VL9SDQ/s1600/vid-0112.png" /></a></div>
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1.6 <a href="https://www.openalgebra.com/2012/11/exponents-and-square-roots.html">Exponents and Square Roots</a><br />
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<a href="https://youtu.be/VhJyEUdBpkk"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEguEcSxyVh4Lk2DJZgFTQAbhtntGL4J5yGddpSg8fLtmkdVS2UOBz_OR2yelQXltgvn-W8nFVbE3hYVeAKLjp8hDLwEeSCweiz6iMbsB08UlBdNNtw4C6RQ1KbtKtNjPc7lb4WoaDGcC9M/s1600/vid-0113.png" /></a><a href="https://youtu.be/6_QE-xrGBXo"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjOZtbGiq1pe_7HrU5DnJeBXqR_wn2AlL1e5f0XWuLTlMkQVHNi3IzXa-4n_XmBJXIGSGGYQn-SXWGl6JaviDkm_0ns-kD9tMsQ_J3vcs8pYLaWvNUsBhgLtYoF5p1aojX_N0RLX40m_Ko/s1600/vid-0114.png" /></a></div>
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1.7 <a href="https://www.openalgebra.com/2012/11/order-of-operations.html">Order of Operations</a><br />
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<a href="https://youtu.be/3pT8gc99M0U"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgbMFgKMi_OW5ZpjLACc8hmnaLTye9nwGcmPjgtuRY2G3nTW41FFjn7GkXVWXXCeJmX2pNPjfzACpM_71txYR8x-w8aa-E3iBeBO4Q1cTmmpw3S5EBRsn3B8c5rHt9RxV0OZnXTQcIJAZI/s1600/vid-0115.png" /></a><a href="https://youtu.be/D4s8YPGQBCI"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDAohQefENZw8kc75zLoYj7VLyUoMXmJjRK_73_6ILDiUbJmo_2GvSbXri8-4ZtL_wQow3Lt4fFdqyVpBZQg4eJFXliUmhp9tABjQCknePBN3wH15jozBxpCJF_Obl5WUMY7dAn-JMB4o/s1600/vid-0116.png" /></a><a href="https://youtu.be/LDwmFzFx8ts"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_3CGZS5-TkoPV7RRMA7h_tKeqGeiDn6rq6_6wHSG0L49BynXPEnI_1mIr4wGat_-uxesrgtn2Bz5rxXHWS9f3sIUxxhZYEgDPDbm9Lu3JxkxBzkCIYXknxX1daWp7BaS8P_TmYVJ9g0w/s1600/vid-0117.png" /></a><a href="https://youtu.be/YGm4QPqDNSc"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhoCTFoizBGessjROqwgPifSEU-enoHarjJ0dP9dfHa4WMaSMQlj7qPYmU3KQk2Fe3Bf4SVdpDOytRlj5Z_cebjAnaEQauByWYsrNVUHm-iN1JHpB1Fa3ONlCF5X7If5VSES-LXWD44m0w/s1600/vid-0118.png" /></a><a href="https://youtu.be/JjvMPHaN7lI"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjRhao3Q2HSniWKSFCTkChiXZeosAP61K72jl-PLXHJmEg4lxd4AGjZlqHDrIiSbtlQCglRxnzBpVOtp-r6P0G0edlodaWyQOcndRIXDgEy8T0QRRzDr4_ZFHKvZOK2bwjlYfOEq2ImUxU/s1600/vid-0119.png" /></a></div>
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2.1 <a href="https://www.openalgebra.com/p/introduction-to-algebra.html">Introduction to Algebra</a><br />
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<a href="https://youtu.be/O7abs8qxlZo"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg38B_XWcozs6pVgMd6bRI3KVdNphBenLSIA63zIbf8NoPay1hJlzx4bsTTYf9KGt3O-gYJouewqbbszI2eZh3P3gacDduNyS64fQGY9souFHaY6g0UDfj-GDkJMN4r6OITWks7DiQIASI/s1600/vid-0201.png" /></a> <a href="https://youtu.be/Pf34Qps_AG4"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgFuV2LlFlaXf90QmiNIMMOxdvxG8VYU2gbGygzD1Mcq6gGI2nXpL8Bp3NtbHvA1HWKLicMcxrM9Fao00j5JASlRGkSh05k8vxXFBSjsjkRsfQZQPTG1luCNmZiZt7_650_MA83XZMI5mI/s1600/vid-0202.png" /></a></div>
2.2 <a href="https://www.openalgebra.com/2012/11/introduction-to-algebra.html">Simplifying Algebraic Expressions</a><br />
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<a href="https://youtu.be/D5eCrBjuQ0g"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiwlsU_kmwB4E_eUn7q-QY0zoX7sEeC3BJpQsiAkPUaNDWJfnsyOl5AUSqosTVPzuwzF3RJ0VVu1632H-5VIXoTn9mJeRcOTz7qS2BWDy-dqKxV9_fhBOntVoJ_34Y9rBbejUlew9pfxHs/s1600/vid-0203.png" /></a> <a href="https://youtu.be/72qA0sCX4zE"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2K6t2RQMF-UuWtrTvNEMKO_yII_YhDUc1PtAOq601BaM4MMH67Z_udOl0XrRHjU7abFXVjW38SUXB2uvlDToyOusuRAMQ8h5ut0qSNWKuQcA30Jdh-TGZeYJgUJ6TF3uqRxHNlhLWbwQ/s1600/vid-0204.png" /></a> <a href="https://youtu.be/pHkEPAvemEw"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjabxJEBMzUBsafhwzE3n0_Yh_XvmELOhX3iSanUu3lzC0yC82m-lKkSlOIZyPZ9Hl-k3fdsY4kaOYpuE0djPbbBJIFvLA9cZ4vxcLXqz2PumGjekHYQE_P-YarCUjzl_vY7I2-yHR4Mx0/s1600/vid-0205.png" /></a></div>
2.3 <a href="https://www.openalgebra.com/2012/11/solving-linear-equations-part-i.html">Linear Equations: Part I</a><br />
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<div style="text-align: center;">
<a href="https://youtu.be/9tHQZw1AmH0"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhOe__5elcZVl27WAQ0A5UR4G_m7KwPZnnrqTjBVnkw9WFvsQoglgLZj-vcYl9xHYhc4OiDGAvD4mTHkT298kzE0kPi3UFheaGKSa1aBQBzl1oOOYcLvuCWrDf0-yj0lem95rHSwSvLt0w/s1600/vid-0206.png" /></a> <a href="https://youtu.be/yBVfAhzQDO0"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiwXpzamVsT0Is1k2dopzN9EXrHu0shhprtlPYFdW87Wp5R6bkhupix3oSR960uzLoxerDc5zNfMv6_Zb1rP20g0pbo2C7pM9JMFEQTVNKL-GnO50ipM4AFuWUweck5_QowV2nNIBP4Y3s/s1600/vid-0207.png" /></a> <a href="https://youtu.be/j8Frl90IY5k"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEitLGusAeTo7wHK9z_ZvcjZt4EVSRiVtKgM4a6okNsiwAMv6Q45oRYqbyi9xTATAuENeDPw-JutrchC-hnr67byrPljXh6ajd7JRbE1oV_pmf-Sex97ZVFr5iFWZ9WlnizJqACEvx4Upls/s1600/vid-0208.png" /></a><a href="https://youtu.be/XYPqjsppBiA"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRQmR1DI7_QXOsS8kFjiJb2lvHpldNkgydl6CUa5mCGvB9y0dAzM_uqcbTFN2xiggcLpxioqjWJ3N5hRbcNLUs8IesiIgChXiTnRLCcrRfK7qcUdu15qCBahFzeQJkPIsW_ompMuIBnKU/s1600/vid-0209.png" /></a> <a href="https://youtu.be/w1x4RpUoGAs"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhT_Y7rmRgd8GpPe9kHmKo6LsnNcZT3m9zi7Mbww0n1Z7RT3FiI-pu6eG97gWP_JObOHw_obM0vUWAJ0dE31BRy_5_MI3G0KOHccijmkfpvv7sSoSLtO2llD-f3mYHpRy1pLoH7C_lG0Ps/s1600/vid-0210.png" /></a></div>
2.4 <a href="https://www.openalgebra.com/2012/11/solving-linear-equations-part-ii.html">Linear Equations: Part II</a><br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/V_8E4161bKI"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-eSinb0BuScRKnsoFsvShQigm9CwUv4mVchxYKLo33jjfdG19p7k7nYXUtZSTg5zB-ceMzisjkkPF2no30shrjAEYlvsJ83GWPvg_6ZINbW4WIyG7dvb2lXC9j5LJ04vFzHjIul6wVE4/s1600/vid-0211.png" /></a> <a href="https://youtu.be/O3SsrkHSm8k"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjsUVEzvgLlU_f46U6SbeFMDM35sl6RLzGTOLNR2LojABEX25MXUaOn9ZPmIXnmPS0QYesGfVpC4BlCFwZsjtGDSn4u7aQU0o-hDo7gR4_sLHjuo9snGL9qQKGUfpR8cEFt5L_15vcbYk0/s1600/vid-0212.png" /></a> <a href="https://youtu.be/Xc4FNveNavc"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg7XfB3hE93Kz-JrtI4wuxGtGSWYS2jzq3f1juGFE19kzYh2uWWW4GtuUzZLjgGRzhTd4UpAFJx9iE65dEsmu4Uo0hWr_DLfstWp7sjal8SDv_IsUnQGzsFeagYKUrVYGlJTVvvi0SPQig/s1600/vid-0213.png" /></a> <a href="https://youtu.be/NNvEMvSJJuo"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjJplOzy_jlKXkjCPkagUI8CTC-k7ggO-_qVNL_FyCLhxmFyXMJT90ae-OTXzhKw0i-l8_w2_43tZ-tz9webyZS28cMTABDgMVa95Hm0ky7AD6aI81fv8iYs4nMqU9bQc8a2cnU7sCpXFw/s1600/vid-0214.png" /></a> <a href="https://youtu.be/ab7AczzKzQE"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiwosLfU-CHZ5Sn7dj0Ggk_nB84py0cW_LJ-zL4h2FuklKZUIVGo6DxXJyKNGhtMqNHpQmsI6LQt1FxYAdhWYYmQWEX8KLzeiETxvEVPvTu7lCXH_PnApuORge37WDabtgFO-M1cvpOBFM/s1600/vid-0215.png" /></a> <a href="https://youtu.be/v7vwjkUAbMo"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEisu3_WsnEqC4-jRNBc-FpjzsZsRoYWF5BWutyY2C9Sq-iLQJWdlWUtBHsDCE7bmsNwFJPBRBx3oNGgJpOCS1jrGacmckQti7lPgE1ya4VqjeO8UneV4wqDubbrPWw8idbWxMo04rqFQZE/s1600/vid-0216.png" /></a> <a href="https://youtu.be/rmm9H5x_3cw"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOJWTqNcM7kOgCHSvjIZqfAYYRewJwpe8p08t5LsTMfSVjkZ9obNDFx4faunYMphGkcsTBFgMV2p89TO8kQAFfAFAzH75CXHkYD3ZMggGduwpaNgMmbAjuV3bi2SCm9Asvt84RyICPucU/s1600/vid-0217.png" /></a></div>
2.5 <a href="https://www.openalgebra.com/2012/11/applications-of-linear-equations.html">Applications of Linear Equations</a><br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/5N3zo-eULIk"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEipCNT_2TwtstQyrQTjm1_pdb9MlJcc-H3zpHyDy6LM4K7TCq9M8DtrsXiZiZLqGiEJwKDQWTLLbkJzuSe5HfEmwN_kYGEKhCYVQeENkBmbii8SdySI28pmKZKW_9N0zvaINVjJ5rQg7BM/s1600/vid-0218.png" /></a> <a href="https://youtu.be/bXLP8VzNy_M"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgcHtU0ZDTlIYwXKRoHtKtFi6BCfxk_2oVJfw1JYyR4sVYpb1K7_6kgD1XPlDpdGgE9rAExUbUYOPbLMFCdjOl9_P7JR4zQ8-h1fWMY0DAfErbufQ4OE3MAbuvGPhGOnl5fgpSm55hMvBA/s1600/vid-0219.png" /></a> <a href="https://youtu.be/WbxOio4aWls"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkVaM6kkn9kUJdi9g2dFUcVIie6cJqFZiCOZYDiNYx99Yx6YVYhbPaa96HC8_spN62cwzL4C7fQ28btVAy1ABprQ1gOVzLxqQ6aPTYlI3PmFBOT3cLOQUavi8jWvqb-erQeVe7jXBBT64/s1600/vid-0220.png" /></a> <a href="https://youtu.be/hRI3UnK0Bxo"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhNcXHOlif5h5_TAZI08hi5UKwkfxGOamfx6PtpOm5PQh-T-NGDlF6jswBXt_aWbtCcTqIBcNZob_1w7GoE6wDR3cawngDhuotH3_gPmxgSteDtZLFv1jgSP8BB4i6WY-lbyptL8zewGD4/s1600/vid-0221.png" /></a></div>
2.6 <a href="https://www.openalgebra.com/2012/11/ratio-and-proportion-applications.html">Ratio and Proportion Applications</a><br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/RAonmW1qcXI"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgsR2ZQcIzDVXJGazNoFM5NSX4We-l2xcNGkt8XnfV-HjF4BUId4LinM1ZWbgAwGuo2juW9tpa4aaP1XSy4qaTN0UFWaK6-EhUe-iirqZJuI-9U0Hl71E_fAk6hyphenhyphenrQVHWbCJFUpB4wfn3k/s1600/vid-0222.png" /></a></div>
2.7 <a href="https://www.openalgebra.com/2012/11/introduction-to-inequalities-and.html">Introduction to Inequalities and Interval Notation</a><br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/97xpuTgjysk"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqwajlHs025aZL_ksGHMCv-_W8Z-K9S3NKA6ZkL_3FBO2U0wa8AN0xnpk9O1NbsTkQzlyA5fwaUur5ZJkJf7DZfycIGpM9RKLo-PSTw2Tk9kSgvYxX4z4QNpnlzjSTfa8DWEZvebmUYx0/s1600/vid-0223.png" /></a> <a href="https://youtu.be/RHwmsKUZo2I"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhL6tEP9nyPJrDK86MrvqJVsc_Gdl4LCR6LI0ZSbKxa8jxvhrY-rYzu69VEg-LNd5pF-6bPKYJQnJBkHk02FDvIVK1Pb5tBjPzoXU5Pw-reN6tX2-hqmC1Ryafu8pmg41e0KyOAvQ9YQOk/s1600/vid-0224.png" /></a></div>
2.8 <a href="https://www.openalgebra.com/2012/11/linear-inequalities-one-variable.html">Linear Inequalities (one variable)</a><br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/KiLcMkPCNOc"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjVukiRxWsZx7CV77S2aqQX_Dybm9Y0iRw5-X0Leab9blsb-1JodRuLqSs6moZh0X9DnOyArSv9kMWDAIVHB5Zbag5MMRU9UdfdLBpNMe9h9j7oMdq8jZZcgtI3fLC_Er8p73E7diVi3niO/s1600/vidCh0201.png" /></a> <a href="https://youtu.be/Mn4PYPpIJ9U"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZ2ilh9T9qlxQb_HTJpU_DpJr7Eh_mla9gm0y4rAv_NJkwSBJSbZVQo_L-zy5ox3CiNvvOBw_FlbuiOqCot6j7fYn0zWXrfqake0rYQh1hmtiO0dG9fJ7TLxM1Uz_aUevhbOYC6HlwIKPQ/s1600/vidCh0202.png" /></a> <a href="https://youtu.be/6kckHllarYM"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiirz6Y2oHjUuC_aOwV5wJvNcDUVfEJRsFeTlrD4DP2pN2THIDIKOKWHw3tnxUjoc_niLiesC-M_ZSuMn5HZQCe39pFKQQujbdgfalKSRgSxJ8o7J1V1infB2ui_BlZ_zxaVpZvDxKKWU-h/s1600/vidCh0203.png" /></a> <a href="https://youtu.be/b-j-btp0lOc"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEitD0q7ipk_kgY8ozy7KkcAIxKgi_EQ40Ia2hyhyphenhyphenmHQL_G5wcunIuIZwSL0DHa7gxa28ZTZR13_paLqK73-W-f94ejRmRHJckM-cC3b97brPWhypL9wlfdzliX18stcQoTtLKaVcsAs0uVq/s1600/vidCh0204.png" /></a> <a href="https://youtu.be/JyH_cmqljh8"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJKOPdAv_So8mVRNA0MdzJZSge-1vr8rhgrWNvybclfm63OVU7V7o4WYMgkJCA2v6MBR54b5R1wQx_8wajIjBlu69IVxP33FzY3OEvjDgJeWYn0MVNNTIJuYP_SqLFanBX1Qnk5XqOua0/s1600/vid-0225.png" /></a> <a href="https://youtu.be/lZmKQnSIdKE"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjJgvr2zO7S2bwCPTLX0ig7JjqWTJNlx1keJwSYCVFYl-HnHoFyhNpcIvicB-rszioj4t_ZvAYKUq-xAmq6cODs5vYVqkMAVY9y6rwrAG1qJCkIE1SqP-8p_Quex2z4NCYSWYt6FG4YinM/s1600/vid-0226.png" /></a> <a href="https://youtu.be/pjpTbzh_2dU"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcktzjC5L2nThEodLd7ZXQc8LTJaJ99m67OPCsg3Fe-D5h0Kdcan70i2OKmCYMlaN5ayn8M9WGgbtIrDspma57r3x2zGku1O2cmSvqCVqKWa3rKVDuV7BWr7k6UsJ2UV7Qq9MfAk2el88/s1600/vid-0227.png" /></a></div>
<br />
<div style="text-align: center;">
<a href="https://youtu.be/yovV0mnRUCc"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgAkSC9cqVwwIGtVV89tMAvqyeb7IA2RTU3WGDCVlUjL-j1TpXwgdkRjXgVQyH7FFvU0mYBTL8QeNgAvld3gD5H7e_JspQWu9GCUZGyx41a3dvt6hzr0x4GF6mLQBF9INfPisI8lqi_J3M/s1600/vid-0228.png" /></a> <a href="https://youtu.be/IOLS5mqWXU8"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPo5nr2N1dyT7ecru0B2ky0pTBx55Gh9FgB6FkMUG_gzdFryAJaT4LY2yZ0DGKbvgJ3-2lOO9uYQYa4wG48df2vaQcFU4UIMy496R6urf3CwcrYhe665cD2h6G-UxrqyPkfPdTIbCFLPQ/s1600/vid-0229.png" /></a></div>
<br />
<div style="text-align: center;">
<a href="https://youtu.be/yzv1JyywmSk"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiM1y0hFpiabbd6fYKg5j9mPnZRaOLOMJP9VIcCZ55wWRrZ6PXulyOxWiIJAjzMEO_gmlonFRQhflVeAq6r2PVTIlKG2EuMZDsZ4iNTpJlfHPk0bOJ55yDnwojBl2FuGva9gHhAYmcTwzk/s1600/vid-0230.png" /></a></div>
<br />
3.1 <a href="https://www.openalgebra.com/2012/11/rectangular-coordinate-system.html">Rectangular Coordinate System</a><br />
<br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/pxhqnLPgOFA"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNM0kOY4kn43xfhyKJT93izIiZS140sQ5LGifflnhCaATA3qM-HXp9nupUUccMSeneq6BThbxIqeP-M51K0oBRkk8IpuDQ23PRGA_DgKHff-FLQJ4iS1Rkn_KeWFVQidECJhAxmzA4tRg/s1600/vid-0301a.PNG" /></a> <a href="https://youtu.be/qPx3dwa9VGg"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDgHDbyAwD-vQt1rT2sDnt9AN_eNvslHo37e0r9f-KEy6t7Pfrg8vz_b_vtUMY3IhA-CFmZWZ-giKtGOFWjuRHa1QsvW9keV0NN9d3C5GulHeoHtwhTLC2WOUeoX-2K6pAoiH4NCS3Mk8/s1600/vid-0301b.PNG" /></a></div>
<br />
3.2 <a href="https://www.openalgebra.com/2012/11/graph-by-plotting-points.html">Graph by Plotting Points</a><br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/PM3ilO64p_Q"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg-3FKwNpNu8nCDtI6KMHgcG7q8ONJUZE7n7WHBaHQBI2emz7z-bxwQgYnII0Kf-0omxNKaaThzFf6tPSR-l0D85lh99m9TDtvnDkb3qHPuJWSulybRSDzoPpVdqIJUG_6Gw-gHe7ZLcsw/s1600/vid-0301.png" /></a> <a href="https://youtu.be/dBAqeXapEm0"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWrCkKm4IeYA_v8fBGOkUt-KERvwNzZs3DgbSpq7t8f3TkXfM3m0kUtaYF9JPq837MpBAd4S1DJdsPqWHrpGrFouzabsgBILnuQ7sxFctlcYURLwfgHb8-86lWSZ3a_X5mkk9cMsTr6nc/s1600/vid-0302.png" /></a> <a href="https://youtu.be/CEhJYN9SK_s"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiVnoFxGIo4U1lh7xxSPaIVPZ1Qf2StONyz7r93dlwa9BmJz3yfpW8sSQx0IhBDXnIwKlTCfOSWcVaeBol16X8jBxh88dJ7-GLzCUodvswRGbmxwu2m9vthFH_MO0P8layqCiOaLa1TTlM/s1600/vid-0303.png" /></a> <a href="https://youtu.be/hudkOQWPgOI"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEis2pxYGJR80SAujfVY0ezrdT6AxC6-NiDb3paKK0ZKbsknu25jHFpdvOSlkHJcvf8_8lJkZYYi2fuXzCTtNfXMwKm8lS-dZ7VtNfficqH4sm3Ux4SDXKS28K1XCQSHtHSxbMil6pdSDm0/s1600/vid-0304.png" /></a> <a href="https://youtu.be/H3mrsuZ_rFI"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgU4Jr3Li8C3xu6VamL6NhpkU8M7P4GsGMCVNpN4UWJTXzElcZ_yJ245G5hjGSbpyXjp2LC2INeXZRRRXLSvK8AKOmA03VNuNTIDCxs_esRSEcWiQppxB03uLDKwcX-obhkf50bBAWIREA/s1600/vid-0305.png" /></a> <a href="https://youtu.be/do5ebh06u24"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiynWVEsN6rVFuUyqJWnGtlQe7tM3YhpqMkoBtxKS7SluM7hGaLLZT1mDfFiA584tkKTZRgQ-5PSpMkPH-0uEycQ2b4i_-mcaZIxkczHvHrVxhlHoWKSRp-UUTuplqw3jpNxFCXu9zrhKU/s1600/vid-0306.png" /></a></div>
<br />
3.3 <a href="https://www.openalgebra.com/2012/11/graph-using-intercepts.html">Graph using Intercepts</a><br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/PNyQDOwiXlo"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEifa1AM5NU6Y29z1ntWvqpU9oWasZexZb5KLxRVnWvpRXtctPoiSqLAbadZb09-32WizsLqBfFfChwbHzuaGri8bc60h2ak8ublEfqSyKswO_sfIPQBgBJrqD3nK_4ohyiQjoyB7X8QNB4/s1600/vid-0307.png" /></a> <a href="https://youtu.be/9vFjvd5QMHI"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgCTMLcjvq2Vl83fA7Zd_okIwQ1amPha1klMzOZhCyxi0qiUK2wLc1Nz-uLBWcMrB_JtgTeKspGRIxlZVOUvBJuNOj0iakZjEoiyok85bYKad07p_UsGnPE1jstLP04WaaPdOflxHqxliA/s1600/vid-0308.png" /></a> <a href="https://youtu.be/fzVYRQTLnLI"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCmjwpZd8sYuBuKRz1yjTfgsfqljfYzfthZn05PXeATvyZpuj7Ef6PiuIDyXpYa3HUXrU05vfC2SoSMSae6kVd-7Nj3iMyxOP53USInxoJpK6zQd3go60s3do8w_Huf6C2ZRmSvDZm0xs/s1600/vid-0309.png" /></a> <a href="https://youtu.be/bFsoJGoU3NU"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEinJDY7VffMnRTi8FiFW5CgEPh5eXzZY3IEp7xruWTxt-aAk-tGMENye8m2rEN1OkJSf4YhGvNmsYaN4cc-ur_6uG7D6pYjMSJAOjR7Y5-cq8fuM5p8JKl-lHO5QqzVMTMQDdW-NivGv5s/s1600/vid-0310.png" /></a> <a href="https://youtu.be/kmsuhQwfhUU"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjdzI5c4wFea55galzlxJdx-0KzM5TTMi8J0zYOSIN17caFa07J7n8hVDyk2PET_f4IzvlFJmT5NvAUHFgGqvPbUxeG8Y42Qveq5iqRZ705LuyNuNxbb4RwfrKOf4NU6E2aGD48AxvqclU/s1600/vid-0311.png" /></a></div>
<br />
3.4 <a href="https://www.openalgebra.com/2012/11/graph-using-y-intercept-and-slope.html">Graph using the y-intercept and Slope</a><br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/Qw2fEfDj-TY"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEghxIml0niwad03qFILFK4S4wbfAGS5pOH_QkQ8Uk8kNgL-Y9lvBb_eusc-a9uJHstvA4LbWCdFUM9LYEr8J1s5_Pzs0biFikk5P5y1ByjEKO1JK-Z2CqzEi_-uRXtFL1ICF980yaKM9M0/s1600/vid-0312.png" /></a> <a href="https://youtu.be/C38oeor_AsI"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxe60ZihuODn1HKy1dDUbmOti4W6HAIqAYG118tJlMfwmj9kH61qH13nH9tnRdjwmdUHQhF701c0U7aM_KiwQMfWWRToCwrRNvo7WdZYKofQ9_6RNrLaP2JLUROksxaMW9NRE70niThMo/s1600/vid-0313.png" /></a> <a href="https://youtu.be/LBP8mjQs2IQ"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNPZC9jYQjlQ_H4wF1y27BW-UJb9JZtIdjw0-dw1mwdgtyGYeae7DRsyjJdRJMjReD8NVETDJPam_Q5MQKLiiBVRHHwrMTvap8xUjM8LUw9CIiQELxSZKlaR8RvZCurEQSAqe6fWFeyhU/s1600/vid-0314.png" /></a> <a href="https://youtu.be/uJlre6BDE2k"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjVSiGiHH5HgRpJBzLY6ZpU9ODdXdxGk6Lalf2qTb4iymP70SKkFEXkw_IBfEAMJ7xFaP6Oil9-mbnguosDimU_9kFm2eCnFeOSdwuX_8ULCKq3sGmREUoya7eqb1n_s_a9ZpD3SUC2A30/s1600/vid-0315.png" /></a> <a href="https://youtu.be/YWXENv1yWSk"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRhitpWJ4Kr-k-SOqiPC1LKI6U_5cJlhCBAjXwnaWihrapOFf8P4jkAm2_BcU30sKOVPoLPD5ASsHiDS4d0g75mKGKBKyJikm8J-ueFwzldvPNdwypyMKyRfe6JgPzNf56JGNDdHzlTQc/s1600/vid-0316.png" /></a> <a href="https://youtu.be/wx1u_zEHa4A"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiaPDXFqL_gc9R7bKMEUr2ZTFd7tK1U-ZtPX37VXpshhUBJ5G6a7Etkv3qyhgGJo0-qGNuJp4iNYJTiK9AuEnTHv0l4-4w4AkK1cf9zqMrkZtKymDItrZFCi_GE0oZsiMriOq_ML-Nrg_4/s1600/vid-0317.png" /></a> <a href="https://youtu.be/cML6vrhoXeo"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgdeT4SNYxT0TzsRVhMgYPlNX6Wumn6AxaQUvnwCSib6MldgKfZxBFBbaITlcdBte2IYdfcX2OUfTgcy2ssjFCSaWb6mAfqCRpeXuDqQ5asaapnj7l6ekwpVL8TH986LK5CDKHFhYhegj8/s1600/vid-0318.png" /></a></div>
<br />
3.5 <a href="https://www.openalgebra.com/2012/11/finding-linear-equations.html">Finding Linear Equations</a><br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/1faPau2DNnQ"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEijvBRjRnbgX1z14W3s_RfQ1-JDPVYdJGRZYGNiS17Ve9noPVLNil85ZY9RTul7Wyfc_I09p7_szazi0r7WYcPZPWgUITdof7c7Jksvowv8zyYUx29GX6NnLfD6rMXA50A7miSxduW3szI/s1600/vid-0319.png" /></a> <a href="https://youtu.be/I2tfXm2A2ME"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWnqyDrYol67KDzpxv-HEjvMW7rGM9kLXVOZOGrehSDr97zfEnTrRDmwDmZXQzLAlSYjm3G9yyBsKU4jfA5j0h89y3a-rFD1nB8HriQDjV3wYAfv-2SC5wEGY_fEo5DSIR0kc_E5R9eXs/s1600/vid-0320.png" /></a></div>
<br />
3.6 <a href="https://www.openalgebra.com/2012/11/parallel-and-perpendicular-lines.html">Parallel and Perpendicular Lines</a><br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/sG0vzzajYRg"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_9IzJr00eORvMJzXDp5ALtimkXe_LFflwK6V0SHYZPQIry7G-qVOn5656_QvNIw2p9ikB7rddMRLPV0F50PfWEM8SJkGzVF800YikWGAHNEoo8_nqit7Crtbpx1Mz5Nn6MxDbSJQ9T4Q/s1600/vid-0321.png" /></a> <a href="https://youtu.be/w_NRKX2YXvA"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEisq2jkn5vBVX3B96zEXEzxm98hlcmlWJbaz_1BL6miukrAsffpIRylvwpV3CImDHxKlcQ847aZGiQQAdP5plUBOYBIXrCJiVLsw6ON3wfCwjWJSNNR1CKePxxIJ99WKa5DUzeYegh8UAc/s1600/vid-0322.png" /></a> <a href="https://youtu.be/83kK0LKubdI"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjz1oFh6Svsc9ilX1IVzSBxgbWhFszWhzE4bHv3ft4zGhh2AzHKAa56v-I4-hounKFOeHfxEzlE2b-7AT37UlF-pGQqJsrqiAfm8hE3B35AO2X-f5aHg7_XDmYOr2-J_o_hvh9tsSPoD9w/s1600/vid-0323.png" /></a> <a href="https://youtu.be/wVgptvpAmLI"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigPqfI29DjHm0XM0F9V4chZK1CPVEYnlH5VlfUHxl2mPNzpjyj49XHRW6aAlHssmONYWd6skFXAft_K1EdiOgxpeo3eIVam2W-W3LXckvoKjs2BX3JmuAVavJUU5Lz1Y2jd5s4xld9EEs/s1600/vid-0324.png" /></a> <a href="https://youtu.be/Gy40rajsZsw"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg94g_NJmV4k1GMshYgaKhFUSu5fON98gIMiK2G2Xn4H7xzNhx6y-ZnqwGnWw9XhxfwiN-f5G_Uc5oM3o4uGj6_pqtwkMlgOPgB5zGIOhQha0o3ytHADrUlaeutn-jBI8TlhGCFXDeKAfI/s1600/vid-0325.png" /></a></div>
<br />
3.7 <a href="https://www.openalgebra.com/2012/11/introduction-to-functions.html">Introduction to Functions</a><br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/LVniBDcYE7E"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEismatW0jAQOScnjWxtyq4A2AYmHxVpNzh8s-LqDc0BZqJOHdyo9hQ3gs2olmexC9mysJiDdKcUQcGhw1i6uPgSp7C1dCVK3u465p7rPcFUDEOvoDo8wZmFd4w_hEDpxv3zLq1Oox3rdkc/s1600/vid-0326.png" /></a> <a href="https://youtu.be/Je0GedsNOPo"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhuTMpkiEsDtgjDQiMkecaN5DEFim2HwRN2poyO7Y1FHFRlqW_9VU0wWA8tzOsKANct625j_Cn90idd91IvJOWAuTAXA0ponmwRCk7o17UTLxOAmJo0o47SqRqT2a3ELGKIJqJqoxH6paI/s1600/vid-0327.png" /></a> <a href="https://youtu.be/zMBSnqmjvZ8"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjqceBlKCmbHhKlwveA5ffhuFX2pFPg8cm1-Z9eo0NL2jFdM63Amq1jxTNiPuDk7LxzMVcFmIFhatoveZlUStUctR6Ot3KCiztme9qCEjvBchx-5xOruvAEato1JIxk6B12V9buifEmW0Y/s1600/vid-0328.png" /></a> <a href="https://youtu.be/qtkEipcqBsA"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjJ9im_GgJ7xJlC48rd3QL30-Jn_I4qK3AhgFKB4ASbjJF1gKq-oNjW7jZ-WxyeTl-qmyxLp_6dABxgeIOO_Byn7J8MfuY6dqcRc8I85TKBAwsPA1OG8diI0msTLd0y9yrVQpMKdUMKaP0/s1600/vid-0329.png" /></a> <a href="https://youtu.be/EQmkiFIW0l0"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyE_TcLJRCGG-Pt68jEC5dqaW89aiksX5D_raM8Rxx8XUvCIF2dbAWatRY1tvBqmV-rTt5AoJSfNiC_RqNqO61iQdpfuX5Wja9KZFja8Fw_r0b1t5jFLa9hsYcwI6O_YJti0lm5S70atQ/s1600/vid-0330.png" /></a> <a href="https://youtu.be/BAi7zXCkVIA"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjlyUJm_VRyILRNS9w4XNsLo4H_MvSWoAoS12y5mMerSyGJFlTBSkJGp3UcN4PDTZ61gEo8mN576cVPWJkrOooqfpm0shl8MVwyna6b82-iO3yE2LvOUKz0b8PYLlk9QzH4iZSgNx4t6Ek/s1600/vid-0331.png" /></a></div>
<br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/fqQqkmHOTc4"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh91sfsXXimtFfLNe0pqeOv9ODI2uZBP3xI90e8Y4twIGUuCZTL9W_bJwSZjeyD9m0AZS1ijjpHx8Q1F-XdVltXUMPPqP2VL58tZ69dJaxO1DbPvNnV6ZrE0KOBjWEORSAQDxM59vedxts/s1600/vid-0332.png" /></a></div>
<a href="https://youtu.be/fqQqkmHOTc4"></a>3.8 <a href="https://www.openalgebra.com/2012/11/linear-inequalities-two-variables.html">Linear Inequalities (Two Variables)</a><br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/SFb-ddo0KUw"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgidGmRnYBd2wsLu3AaQyAGh9SEUrck6HExW3gxQOU5hUl3cWnkf_RVBmBcsAxz_Lk59ygL9dMscAPuz_MVYCCIzGi4Iks-MKct3itUKmcFNPbZ5fnu94cZoNKVWTqACAAJBtuu78vtj70/s1600/vid-0333.png" /></a> <a href="https://youtu.be/JkrwnXipUKk"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqg0t16UdRaSHtVijcpONVviORr62RPSZntrDAer-NA3Rb6pvY9FNNR65vKGEnmDw6bgNUlM30rnyotIsQukRnYGYljymyWu3-lfi41vrpwF-m1x5O1lslR0rGvDjhHM5hCZ8CntVgMCA/s1600/vid-0334.png" /></a> <a href="https://youtu.be/2yI5QjYeLuw"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvStL_Euxfasxr_aKswlRyMPta5IE5ogg8EuymTr3C-kzJwN9QvvgjWefuUix77YEc6vSK8jA7YQ3-co7qXTtb8Af5s5DAnPyKCed0X0_y1Qd8MFg7082RmjF7XSvVisqgYRZOtoFmVA8/s1600/vid-0335.png" /></a> <a href="https://youtu.be/NJn7tgQugyE"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjPWT59UUJrMOj2MVzRs8nMKxxu0kOq7SQAZQQIUcuFKRsM-E4WGSJzDfEuZX6mk91Uo5wk6MkFuOssix4D4CwtnCVk0lVYqu-ax1_gUPgZ7R1RaT_QZPl8ZsNGOqeTAsAwdj0Y5clloIU/s1600/vid-0336.png" /></a> <a href="https://youtu.be/0yLLPpsoFWA"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiiTSvGxn5fYDKVS5F1s63CnDZSyGKfmPZ4KHoWn1tW6Dpi2nugj54O-KAwKT3OeZmRmXLvhO_JGqQ265CEXttmYpELHulT0RS2_mPnvLd3Hc3AQP1LZlpmQrUK-FEz1wDqOtlxvo_US0Y/s1600/vid-0337.png" /></a> <a href="https://youtu.be/YcVJJuAA2cA"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhyqFQ6P_7YBo6GRmFYy1I9y0W4L50Sq8PtbeSozwfB-rD7ChyDGONC_c380tWaQLk5rNCRbNUV2Noy0sOdoDGmxz7Rtoszx2PaGrblhKFQOcMB7PR_Tj9k_IQDef_UIUa51Enhq8xlaSk/s1600/vid-0338.png" /></a></div>
<br />
4.1 <a href="https://www.openalgebra.com/2012/11/solving-linear-systems-by-graphing.html">Solving Linear Systems by Graphing</a><br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/wuE_KDvjCPc"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjqsxNDpdYt8Dub_9Wf69_tjKyGpPiOMpzQeeZ8GlEAMaVG44nUILWvm2T7FVDAR4AYFzTpPyiK6obamBduqii7kYrTdDw89RC4A28T5_dpuAZbEZvvk6R3I3G8NAe2VQor0zbpCywRJDU/s1600/vid-0401.png" /></a> <a href="https://youtu.be/jtwDNc_AIew"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhBKRuJcItM2x1SPnFsjlhAsW8QPvwVXIVz1WeGTUDwnfEgNCLF2d-jE1zpfMg3S0J45-Z1_PP7Itpk31luOTdiKHaoKMUTr7GeiYCGgec-7Bep9QiZG7cFPgQ3Xp2pqugoArMf0ZcZIdw/s1600/vid-0402.png" /></a> <a href="https://youtu.be/CRShq5LIlTo"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6AWwSiA8yVi6CbUYhkrzUcWpbi5Oq1XWv4dM1kqUoNVn_eeD5wmEH2-pCkvHIElYuQEw2Zhgh3QW3Z-wjhv-jFXkBDhmO1JuS_AmwcZRFsZwOiGmJs2SsYo-vXg2pbgiAqKUA6s2qdgg/s1600/vid-0403.png" /></a> <a href="https://youtu.be/IPIN6xmmhs0"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjR5a3ZwX13OYExsTV-Ux3SQmIajUvDeU3qc3MwAueZFQEsQMZK5X8fb8P5FXFY5qqycJbyCGpzJ6b90hSWmSLAJ84stDnvV9B2f6R2Bh4j1KYhJS7J9g22BNsux8bvYG58_E5Lo7-Eyn8/s1600/vid-0404.png" /></a> <a href="https://youtu.be/zla_JLLBSBE"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi4hbcnSR37lNOt6YEuYpY3jN7A4-qGWGyfSUeHy-rxjP_fhbBJnXdWKFKLpSHirWI7pmtKBjWfQ7oDJKuwjbMjUMd3RgA5AZdXeYjS-abrpG6Pq1PwUI4dX7FDo0fmHbK1QEv09Df1Skc/s1600/vid-0405.png" /></a></div>
<br />
4.2 <a href="https://www.openalgebra.com/2012/11/solving-linear-systems-by-substitution.html">Solving Linear Systems by Substitution</a><br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/wxVheBIJ_l4"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhDB7PHGpvi9prZdMgDkEcyPXKWZlQXRUxR8VxjA-7yAuooZdGQeAm9YfJID7hY_CWaX0-X98VZAJCGt-w9gh4D_LaZ0fHIQVkHjYbKmwtB4G8jIefi-ag5y8QAi_rmKV7Ze1EOZH0isBU/s1600/vid-0406.png" /></a> <a href="https://youtu.be/lXeC_Oai-C8"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjjOBE1tv-tBWyFwfrKqTMefxC3iNIdE05sOpklf2c9aDrbWlWU1Rab6ufFOLR91tuUAcQK69q17hIDWt4DAc-kVYMamShLnPhiN0boSA5LaI130GAWHFuyGfnzQI1Te6Utq4M1_Z_6ZeE/s1600/vid-0407.png" /></a> <a href="https://youtu.be/NQDhMiB5NCY"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgbRFX5NowiIy3utn-SyHztvq5unXIjtfHBYHZ7g5qKW5SsA6pv95BMXnNf7Ah0cfQrl-npuy_bkRyXkSB4a_d7Qy7CctdSH2brMGLNtKt6Wc4qXJ-t0PRxHhNeGy0osbj5MCPHwoKy1QU/s1600/vid-0408.png" /></a> <a href="https://youtu.be/hKpqjRz45Yo"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEitYVjRvd3r1CsM4gVCSpqr1XMPeJYllzFF_JHrtxp2G3Zf9DzYcnc-Ppnsn8-bzHWTi68-Zz-HRuNxPiM4yjFQuB4Z6Wy-b5rWcLKL0klEdFEgUVYV5Sil1TeqGfVhewKiByDu8_2DkLk/s1600/vid-0409.png" /></a> <a href="https://youtu.be/gwipji0r2rE"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgxLgy-nBdIGQ3LneI3RfuqHz8IpE4UVP-r-B6wa6MFn-BcmjQyUzmp8ptjWtJDxDeiOlOqb98bHiemloTgjdasAp62PCX_sFsTqMEUCk3QSJOM2dhysXjemgrJ45oCQEflHdi-EcvMxeY/s1600/vid-0410.png" /></a></div>
<br />
4.3 <a href="https://www.openalgebra.com/2012/11/solving-linear-systems-by-elimination.html">Solving Linear Systems by Elimination</a><br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/5xxH5MEFMEU"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjBe6kFRbZQK-xz5mFAw2AOrJMnP5CLAoV-FPGlnhmYaGnlEPlgvLZEvv9oyvbd-e9RThsIjCcL3THiQRYF45SXGRmzQCDTVpa1kafkzjylm21P-SCg4RUSvosuTe8kgVYmpxNFK3nRy70/s1600/vid-0411.png" /></a> <a href="https://youtu.be/6boqPMg5A9s"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhdRcqnyGAcLyTJdikwECC1RolWfQrtBBtMWqLbU3c-wV8CaM-SuLgZ_tzEjGlQuQhMsLTCF0swuGm-OAB2Aj-s4KCtDU6GAQ4rlxKyU3vQPGTFDLpircuWodRp-BdrQ8u5TdYoX3zuhnM/s1600/vid-0412.png" /></a> <a href="https://youtu.be/mgk33nRoU58"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhSwSlfekHuLRNdhSc00xBZGpUraR8ReeLoXlP9sw-WHawfXU1EVty6Z11PVHhaNLPR0uZ2HI7JCTWJYyrH8uTs-V5zGjylXKV5Udb-wovcgvqkCcfi0FZTcolUJBCDIaDExq2MPh8oL3M/s1600/vid-0413.png" /></a> <a href="https://youtu.be/tr2X4X6r2V0"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYYxROhHSoqipkRJZUNr85EXgccCmIfG-G45MWJbjKLW_sjkyTHNcXywxomaoIEKm9qwsysC3W093HigidaVo9YoxV2SVnZwIA9wSLLLJFQvfu6RFo74XqSvtuBm1ghEm9_xW2Y4m_FaE/s1600/vid-0414.png" /></a> <a href="https://youtu.be/HO0k_uvp7Dg"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjTYsRzLWvL-zYq8BqcCMTm9QR7rodTmJNo5bySYPa32Yj6_wl3RBmoVbbEjaSBbghDZwHL2GoowEPo4IRVFaQCKJcEpW-DzWrvY7_FukXmPGom1ZvIX5VOK1wJcis1255veg9ueuMPEzY/s1600/vid-0415.png" /></a></div>
<br />
4.4 <a href="https://www.openalgebra.com/2012/11/applications-of-linear-systems.html">Applications of Linear Systems</a><br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/yS99EdvqYqk"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhFLW2lm7OP293yQ8JFZMJSLObpHvjnkbDtxSlUJzXhDE2W61dv5VKoJJstEJc5kxTxMoJDrEKJ-iiUQqhiltyYcVLGFVBTDvZ2dI5SwjzJia49J4wmXLQzsqF-nZc-HKJUtD7itYimRko/s1600/vid-0416.png" /></a> <a href="https://youtu.be/D3luluiPbKY"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjh5XieNGi1Cgmc8F5pGOx7So2LFhFvaJPGvEB3ij1KPlVpYXgPWYaeSnfGqVD1oMh-x2r_bP8ZVI0ExCykp1_UOE65t6hRKyI2zsYgSKoiCnCt1pb8IRFRFJR6lWgBbSv1mqq8GA9YQNA/s1600/vid-0417.png" /></a></div>
<br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/JZaTT_m6sSQ"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjx4j36QDOUPpj2ceHx4dkhVYhp_6hKRI_2dXtO4TSxs_QFqHJo7n9KoSwql3pShyphenhyphenWj0DB0dc8YEMvDWfAjrALSn-24Yy9eVVpoz_WKQPgMCxKxesJx4LlJXAj_HeTqxds3hsc5KlWo_RU/s1600/vid-0418.png" /></a> <a href="https://youtu.be/TP73Jx1bADg"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEio_G59sPIltbZ1jfMnSpPA24r40_EgSLShuMMWdEkxm-Xggo_qYIqmL0Q-2BRKQ5AEmdrjXYNe8poCDQAWD4VhzcaZ0Jmst_X8ZYx2p9l63W3FwKb0ooFyvuiTsgmT-Qz9FgEwuRBO32g/s1600/vid-0419.png" /></a> <a href="https://youtu.be/238PzrPC3VM"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi0HhydPuo3Lou-0N97R9DNYkr26zvh7QseaAQtsBRnJgEc_4iBdMM7g3etGyeupG7fYTzPdgmQq5kHeH_aQAg9D-nvGNjfFAACjw3oUZg_zALrRX89y6rG8fg-Y8P9r_AW2hLMUnXbuqE/s1600/vid-0420.png" /></a></div>
<br />
4.5 <a href="https://www.openalgebra.com/2012/11/systems-of-linear-inequalities-two.html">Solving Systems of Linear Inequalities (Two Variables)</a><br />
<br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/uZHcyib9fjY"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiy6iQ2sDnEbpD2vZLTWFV74pkAMOsdLG960khr_PgK9pwCA5clGoGYxnnzipTRTX3_VhMnTFgXwRMZpGJTT_irFHpltAL7VTcdxoRwsCbFh7gCibkJecNMpmsv4ZcEad7n0alVfwuCnec/s1600/vid-0421.png" /></a> <a href="https://youtu.be/RWnf0Oobzgw"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjqnrGdxOVgZG5uebxg_R8q3rEswhWrB2Ldoib0ujr7qoXPtQY-U_UX1Xz_nWXc058Z87hTTr6f5r9g5k0XPtucOtVxBwFrMGo4xtthB66ybBhkjBB95WPTwsuwNpchrGOgJKjH28BxLBU/s1600/vid-0422.png" /></a> <a href="https://youtu.be/wZMOI6_EbUo"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjc856zg1Jta7IR_ndrw3PSl40S69B56np-aKiYjp67jF9-ayWYjAYPKIa6XloNcr4l3KOfbPEojWEFdVEX3r9dSHdp5Y0osQk4mj_rp2Ox5_2dIDdVCjPF8fwta9pbKts8HHq0mDgP8II/s1600/vid-0423.png" /></a> <a href="https://youtu.be/SMk6n0CG8J8"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEht0kjnE_gfZ8wFticYlfRiAPkAaWH20PBAFlQR8Stt4hU0PTuKEpnMN_HlfIIwtY1F1Og6yMqV2eYdawNcWKpm0UC9jUbaDP647vJUDEvlkQI1yVzWtGXiocfbnLRVvRXde5OFqp-hrbc/s1600/vid-0424.png" /></a></div>
<br />
5.1 <a href="https://www.openalgebra.com/2012/11/rules-of-exponents.html">Rules of Exponents (Integer Exponents)</a><br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/93zp4oCzhUU"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEheIxO8JIQ6kLliWVh_NLrVvCdv4TCREb8XxyatA3tttaBwH7GwqQVBapcBkqSaydn-kXS8RaFtfCG09SrKMU-s7Rtv_EzXTDVAS_BsftRGARBImEQqyhLSV8_ZlwqxDRXpyuEPbXboaW8/s1600/vid-0501.png" /></a><a href="https://youtu.be/m-c7cw0HNWI"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjiD3skiBHD4WdjyQ_6z-MLscQjkR8nmr_zmRAEFj3HvjEy7j3Vpgorg4hzuElNydaGBQxuRZBFSn46ecQ1jxlF1TlErxIWEXRPkcmORyEIuFIjREBiWU2AE_1BaJwdzJHXkduoaM2FyEc/s1600/vid-0502.png" /></a><a href="https://youtu.be/k3rZ-NG4jEU"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUm935SwV6ewyw1EMQI8fT5QeW9aSH71hWPYPXn3GuHUsr00GSB9ze2ZAWt9J_ck4IpBsQADqHWA1p2eetyHbTP5f7AFKtEx9R6fU4ms8MqHYNqaMrFF-yDN0VH3-3DjMVxPy0KjNv6u0/s1600/vid-0503.png" /></a><a href="https://youtu.be/hSQRaek1lLI"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEihdNPD9NSEvo6XmjKRgn12516alcuy9cEyrxVJUcPdYD0TynOTHl36pghbArPCL2LAdmqkFBofUSKoLNzKwSItcUxh6gjLh2vI3LmtvNGb-Eaq1HOaHH51Tge5Q9edJb_5WUV3UqcQj3o/s1600/vid-0504.png" /></a><a href="https://youtu.be/eAFB3Yp3SZU"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDu-yfGIlI_PuXfLRQQrk1M9epkvPZ-HwlW_016A2pKCFvvpsH_rR6RfBj0MMJF7yRA_6mTU-0Tj5RFt1XWgAkiTt8P3VyWRPZih5A4ucve-HM7EQGG070dFnLGnGREDLrB-z0CMP3eOk/s1600/vid-0505.png" /></a><a href="https://youtu.be/d28tOzWHkWM"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgQDpWmOMfnrPaI_42sjEdRbfNhS0QuATf2RUgn_zfsXzqug3GNZ_0Usgt5rBCoC_66uCbzLJK8J3GdRidTCtTHS3Rl0x2YNNvQblGu1dhgNWD0MLU5HAmQkMgSexL-lrXx17Uu4eSHXRA/s1600/vid-0506.png" /></a><a href="https://youtu.be/SxfKV786zSE"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi1c8XWC2xfEqmEhUVXrxGiidLtwuAL_33cgT7826EAsCF4OhfviZsBerVlLFDvCLLuOO-QXap9fNRy8Ihc1kAc0EKLaNBooWQZILyfspEF9SPiwOKB5LowtfdpesHpz0yh3fichp6gvFI/s1600/vid-0507.png" /></a><a href="https://youtu.be/Pa5bAltZRoI"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiccDXagWSRyfq8pE0w4KCYm7cXO-c5lgSr6hyphenhyphensY3PERi-UhHjtUoG_0eV-HAcTqZNgHcimxcyiIFU0G8p8pmAGhSnojv_4Ad6noVky6H5Ej5-LHRSkzJjmjvuay7aqkqn_g5qW7cY3rfA/s1600/vid-0508.png" /></a></div>
<br />
5.2 <a href="https://www.openalgebra.com/2012/11/introduction-to-polynomials.html">Introduction to Polynomials and Evaluating</a><br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/Ihr37DEU7Zw"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhFTO17yP2xfT9BCyysvRBmIJyM1395b7WazF5wr5PIH3yDtm7coe_7VmEOn_Z-rhH5z0upM2r75f9iqs1a-1KK_KK2KqhXwCQ71-tRoy0qJg6OOtLkYS3SYqd_oq11iKudJA4mzlHgbhU/s1600/vid-0509.png" /></a><a href="https://youtu.be/0Oj4Ze78gUE"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiEjR3EDfQlVElPQPEM6bLhwHgd5oKKbZH28sKTOx7cfvXKYYO2-YjmgiU-s5u4oitz54wVRqNS4_JjD_5qaJAAXwtQEZiG8m9QuGQ03Gzeg-5e4clqNdUg0BvSXZCr-9OXZFvb46R_o3A/s1600/vid-0510.png" /></a><a href="https://youtu.be/romuY1HErbE"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi1oAgmFfZwzVTBOB4864z0g-81vl_c2XcEdYWcg_E716jE4Eu-Y7IJ6dZRTN3CgMgEkUUuRdHOxSJuFuQGPF603UFtTHQvnO5ZsGl5YxZpKcu1lRIAEVr43hsDPjna8L6fFCGDh_Tvk-M/s1600/vid-0511.png" /></a><a href="https://youtu.be/orH827ksszI"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjc8Iu8dCaYlCiKN16AqF2jcyspcdVotMYMdTHjHn9UdOJSouYdjNdARsVQb0c1L6uNEa755wzK1wbuvEczXkvqVHm-Z2RF8xtMAGEPJMle2Apmb93I4Qaw97mMBj2f-zYTN9fn0GnW5uM/s1600/vid-0512.png" /></a></div>
<br />
5.3 <a href="https://www.openalgebra.com/2012/11/adding-and-subtracting-polynomials.html">Adding and Subtracting Polynomials</a><br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/16RMshkh4M0"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJK9bUAWpPTYtMMj4uAQwR4zMFC3JpXQ0uRoKU1T_vNjEbL3cRyiUUTrhWDZK4Vc9a0XPKs1zFutdoHMiTtqXPk8lmD_70YzgL6J6p2h9YOgWbzJVfhYh9dmWWcUhzCwGjphdq_drsKPk/s1600/vid-0513.png" /></a><a href="https://youtu.be/yvX9JVzb4eo"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiGFet3HToqp-ZbeKPO-zNqzV5bNapuRR2fqbGGL92FprvYx6trBG23PLGmjsJgvZvZ09qImeH1_nQbc-TLITG1iqCgqjueV_qMJ7mjiAbsfo1H_CcV1BoTo76NTZbkyywLbpS0SqSCUfo/s1600/vid-0514.png" /></a><a href="https://youtu.be/7NlzvyZfwkE"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg4hlU806E9djkCIxC_U3PeRjQZlnOGU6UEsSVdx6V8oQ7LBMjD4s1QJkOxJtevm8-1u0r1m93-lHvnnESPT_Rex2gtavCkHM23rjSmoYavh8oTNI8ffZOXwe3WP8CBjZm33CKAFa8Pc4k/s1600/vid-0515.png" /></a><a href="https://youtu.be/QihkxByoR7M"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjzmYXjJLVYoW6dIsv27554-D9ThjbVJnLdSoMQP_1ot-1rLscg4uMPfPAokV05qGhyphenhyphenQB_M-0Dn2MvchuCS5o1PUBcryuYQ-DqCA2Qh9dP6tmrYOuy-gA2XVFx1Ykrhf1-mtcrUFbNYVyU/s1600/vid-0516.png" /></a><a href="https://youtu.be/dEZzng90wc0"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh0DzthVXjXGkwfY-Ww_ZYUvG7zghBsNJ9l9CHPs60J3NaZRLHCcG1Md3k8RPTNaLaIOzZNmR434MxFXOl4GB2Roo604TKyvRj612gpQRjKYgXI8QB_bWp67IzXMPzEvaW4i5bKY-cdnRM/s1600/vid-0517.png" /></a><a href="https://youtu.be/_xwWWoQq7fI"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjwg2NPwtaDOlRaBBjjzyGkZ9fPIFnrvGRMOSrZJBikIGmvrwn7WmaxX8tT0vLzh-i8bZcp27iCBwfUw6GY1FcY0OHN_2yjJPf_TOowO6HIZukze4MpaqnOXdjhnVGi9xXGwk53_HFmW-M/s1600/vid-0518.png" /></a><a href="https://youtu.be/m17GQA-Pi2U"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjXXa7PyxJlz6UW6gL3OWmqDF5xGaZMZ3xz33sS7U2pGAFDmFGLGm0S-l85e-QxX8_LZ2UXtHT8sZCjgqzoX3NIbxsrFms0jwPOR9U5D-ab7_FuhTW5nTVlv3c-vEjXhcG7cc0x4SrdYcU/s1600/vid-0519.png" /></a><a href="https://youtu.be/URT5pIwKJpY"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhOnrMyPJBJIa61yIHDk76n72ZVXcTEnY4Lal0GFCC4Jo4Bhl-gfKriTMkLsD6EjIaeNX69C0MoePiC-CTVP6CL-0fPbicvmftsgWLyXkgQhaqOCbId2NlhMB_fSif3emYYizb-7XIpIWw/s1600/vid-0520.png" /></a><a href="https://youtu.be/_B4Dx4mIFJI"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjJQ1VvmzqAF2Nu4_VgG8scQFbv49egd2kcslKbqwgcz1v-3Gmkv0ykR_Eia2625bBzRW2kTuqS-_ZQr2LY83BJxQlLKTXh136qGlKpgiitxEL3esAvlJ5nR1wk8NvBHeEE1gBE-0I1ub8/s1600/vid-0521.png" /></a></div>
<br />
5.4 <a href="https://www.openalgebra.com/2012/11/multiplying-polynomials.html">Multiplying Polynomials and Special Products</a><br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/Z-0GLPLKFZY"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhNlU7NqObGsXCrhKaPl9KhvUCOe5_k5NpKqdBvw9D58djYITUU4woVHdvD-p7K4B2fRu4LT5V0czLJAXDlSF04UrkNPqcS5-WeI1tDO9cVdQ4qTR_RGnVx7OXZIXKiD_Q8x1OPmU_bu30/s1600/vid-0522.png" /></a><a href="https://youtu.be/xp6aTFhZ-Po"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjEcgbLSW4R1nczbE1reQwV2jPMRFxkDE8smDLANbkc6QInQ6o0JWdTH844e1XKqx_FX7XazPSw1qArkWDn6loiebrFhIXDcAAx-tUD7aJNou22NDVI-wEebDzthaQYxghexjeTECSPvds/s1600/vid-0523.png" /></a><a href="https://youtu.be/CiBBr4nLpRY"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgF5_mEKOkb7Oj6tSmXAoJKUBy3LAllnNLuGmEDN8cV88_57rqOT67aFkZCKwuV4X8pEq7W7DlGY5y-X9_xvTclAdO52JQ_lyJcvdjmc5rg7Otx5WT_CPYSoczFJU6C797lytmvzpKGrz4/s1600/vid-0524.png" /></a><a href="https://youtu.be/WNxEtjlERsI"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjkan3WS2BE8lp9FnMDl6EZew6poPaDagqsW35O60qkeCO-XQiY1HAwZLdOqosHUihdpMYsiJ-uoccTCTtZ4Lj83Zn1B_eJyNgxylxRUaydOWFfUuCMVrPJFcKR5TZ8LNfwYOjEtGBqdYE/s1600/vid-0525.png" /></a><a href="https://youtu.be/XXdTl5ZJkvA"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh0xJqGCVumS0S-Oij7opkKShxZlkNKeazEZDqm0s3VGpmpX7DtQP-090rosFHT_q8fs_dHQiOxz26sScboxlfwCTCNhGVqOwpTp_zZNoSF4J5lOWT3XY22h4gxPaErXzJ4Mmgoeq7ecxo/s1600/vid-0526.png" /></a><a href="https://youtu.be/02gIEfrnmK8"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjf1cckYxoWzByPoIogat9gKVNqKTZLlY4TQIb5joVRPpm7kAkVQ14RtZv80ho_9DU8xB9md8_RG81Hfdl8MDn3OvQjGIFx8BHhMR7owWDnwOX9cYsgLVOZBFlkPWBtlLzPc5BnEz71SRc/s1600/vid-0527.png" /></a><a href="https://youtu.be/lXq1zM4aUZM"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2nfjre0yJTFP7mJ104T6O-HQ9NXhxLizdNU1LpqSZzdO_zAG-Eq5A8XkPJZWOeBo9Nbh0sP1IjnBBhwDdUjZ8bZvvpJC_HulZpZ6p8d2RQS7Z3I9AL2AYFVGeoyAxRV3jVNBy83cbj-g/s1600/vid-0528.png" /></a><a href="https://youtu.be/Q52r9McCkHg"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhRuIlz4i7dCkCtHwRILJ73MGQBCFnIylrNn8bmblf2YdYpeqHM36Y-oTVQXi4ogZbzjcrStJbz6V_OCvJDyp2mAqtsGz9wO5BIysP2RszMZRZZDl55ijr9sCBWpaho2BMm6Q8BellcHMA/s1600/vid-0529.png" /></a><a href="https://youtu.be/jmm7J0L5ffM"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEinmbBioOCxQ-wvA6DILT3_KEjR6kG89O335Q0G6nbn_n0rMvE0BRE_jhnZavmkkPIvgNBLGZMgrHMxAzqdvwmH7Pwr_T8zVZUD1WHzm-cqzQWnKEE2YcRjQC2xJTFRe-gchTdchTcl4GI/s1600/vid-0530.png" /></a></div>
<br />
5.5 <a href="https://www.openalgebra.com/2012/11/dividing-polynomials.html">Dividing Polynomials</a><br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/spwgEKC1UhU"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgmyFQYMDtjviw8fpIDne0JmuBTW1ptFiGaFWAYBApmSyzuYuwOVnEx_k_ue8S17mGMDwzYSHwBoUorEcmxUWiI0qRAHkcTiW_SxZDV5AwhVjC2HaEDOgGKFfwvNkaX4tEb_jq9CCEfEk4/s1600/vid-0531.png" /></a><a href="https://youtu.be/UtIwkP0RXsE"> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPALBQdeo63m5VvNn0P2hgJLNK80zfem66EB0Ke88QCIrx3Jj-Nn1CFKMRIO3V_8nt0cS-vOLxrnlRZYARSOYDxueQ7Nip4dG40vx1yX9zDHENPNSeqGJ11BhEaSc0_CT2kmymK0N0Oa8/s1600/vid-0532.png" /></a><a href="https://youtu.be/G3_ZMWm3vq8"> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiL0Ultt5O9jXh5D81rdFcdaWZIfa8tkh7snRoSaLGYYwMXFHBX5NQhabu2LHn8vgleG588xVOlyjcZTMaaZXD3gKp9lVQbNO4ED59HXRvsrLnnaV2qyTX1MXb7kJ1401YR3A02OBFgCNk/s1600/vid-0533.png" /></a><a href="https://youtu.be/n9FQ_oQDHJc"> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEikBu2woRnr7yio823uzkdpDoSdvqJbadg7GiFOq8KXJxI_PurtbRoGriWVjnVP_64dVaQhAertpONFqJUwe9ywotDB3Xe6XN_U7zIH-lZXIyV4KYZOZ0I18fOCBNVMf7HrxrCWFZWPw4M/s1600/vid-0534.png" /></a><a href="https://youtu.be/M1rDJ9Inb5M"> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_rV74EaePQkwd6euDSsaFmP09voyTaL6GHPRLgPtcV1xK7KsEtmB0sZsqqrddikehp2i2wCyH8LNIF5S2IKzvtLvx2R4LL4EaEJiIoIXrTNvKE43rFiniOwFS74mS6x5WKdAZhe6J0qs/s1600/vid-0535.png" /></a><a href="https://youtu.be/ck69C1_S9-s"> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgB0lZl7jBAFNr3Na3y9UQHNWF_RGabY_76aurKmqXE3Pv0GmS8NyNi4G_GRMe9g2AKdmBZ1nKaccE8-ZFBSzJNNCFwimgW4q1Yn9i9pcGvxd70oZkJKWiGnEz92xWy9qPjfmJ8b-T2Ihw/s1600/vid-0536.png" /></a><a href="https://youtu.be/QLYdGEMDRh4"> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEipDZA0zzDR79xNjmd_tKOrQ-2Yt4nnYO3o4dWQ2_XVbu2hiG9XCBtTu4u6a-bQm_NZG7gwGtzw_XcYde5UiEPM4ASkheP089Dw-cPVIsXsgVYSU9_jOMdBnugEa1ev5tzwgrFHSkneEHg/s1600/vid-0537.png" /></a><a href="https://youtu.be/alFZxCnN5aI"> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjgtE0yTePMYh8KzrreeXUm-wk2KvZO-QQXQ5Lck2uhlssTZ4dZJ47Bog134vDSjW6eYh1kzZSEkjDiJv6yRyR8z5WF-3tENR2zSyM6Z0oVQHZ-_wdRmbML6Dd6eNOj-m_Y0I__tsDgqCI/s1600/vid-0538.png" /></a><a href="https://youtu.be/afat9BVOEhg"> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjG-0eRP-qY9gQvMM0esmT6T2LUTwLExyecaMSSdsaCea8JqIHDvWV8EQPM6KRgFwYgV1gjzdOafqByXWZs2gbJzI6sdvkhxZCqcP89D2wNN7BLFm4J8XCRd7WHeHIYbYoKpirxLava7B0/s1600/vid-0539.png" /></a><a href="https://youtu.be/p_SgOnnEBBk"> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhbh1JWUq1sGqYDEJ0WrkS8vzPfNFlISQvj3GnylOXIwx4nC6yiM8XQoRNm3dXFqRCM0AFfB_93F8CoL6_KQk2aLo4tmzgOjtHLOFxFDn2Aov41B55YWqp8Satx0bpUFD6I6MLW1Gu2nW0/s1600/vid-0540.png" /></a></div>
<br />
5.6 <a href="https://www.openalgebra.com/2012/11/negative-exponents.html">Negative Exponents and Scientific Notation</a><br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/60xDecdX5rA"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjs7XvInpoNjph1NhYLmTEMsyxCx6E0g07SHF59ZUGEx3-y_zz2HVZ284xjPVJ5ezgXUd4LFz5jdc28q2-6LaXz0xehRUui-9bQrSrztBCOGmcrMuEQIFNsbgZG-9VlgyMSXZaJotG7tE4/s1600/vid-0541.png" /></a><a href="https://youtu.be/wRy_SwrPDN0"> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjmlKB0XdFRtdVspgCLtkupuSV8d3of9f1uPSwv_66Yrq6kYSRBHbqaemzjtROR_rsUf_uJFYT81zvhZRJqWcNRG3ziFA7Q8sz7fGpwMZ3Y7drnyaKQpjKXpL9zChBKy36ISyzh3pPOMyg/s1600/vid-0542.png" /></a><a href="https://youtu.be/CxzNVu6ODvw"> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhTs1svCZtcV4Ars1KcvNy7dWkvz5vTkp5oR0ZzvN82DVXrv0fD-nJBFpXUnOHBr7EvYvs0ZinrCg5mNlslu22O9aNRTsk6H2KlpTk2gT4IZK6M8PXW-aLBztRNPgtT0SpHN6w0WoFH1ak/s1600/vid-0543.png" /></a><a href="https://youtu.be/HT8CqqaVKZI"> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXHENiki8I3pYm7sUCGniAS-TE7204AclgmPYJYP8pF8zRD5TLHgg_PEP8AxEgRqF6CJaiW74A7yN18kVI_GyIjvGxSuhcdQRr-MyU-t8PYqDrEOhieRd4pAScHqenyLwKJg8GvRM4KFE/s1600/vid-0544.png" /></a><a href="https://youtu.be/p_ILSeuuvP8"> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhquTaLEB6wRCsRv_rpGQNlfJ3IcEXshy1Cnmt7N5ZtX4TW0Xu4uicl9dIBEikDaNweCEmbUYx_0lrvav7hF8kLDnzPul3roOZ5ONXs8BZFEoCh4p8Tgx4zjPhrDmzJNfuvdOvo4pX2gTI/s1600/vid-0545.png" /></a><a href="https://youtu.be/bppFgsRiE7E"> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjXzVZ7lBSPvigHMzTSKxrrJAAJvP6CmpOTcp4-dNY2oR4bNIOb3ccmP7rMdjGpvxorA-AN_YooG5CXDs-K2vQ05SdCQQJS9JadOsjHy6H_1Rs948_DqOqZMyyDoEkBZuHLaCRbQHybXfk/s1600/vid-0546.png" /></a><a href="https://youtu.be/zfFdw54WPEk"> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhbvRPlioI9HsXIQoD_AzYPyuvaoG-K0fE8l403_MftSxfOQnASQaSnif0Vbfn6BUJd_SamlkhGLpzIvZrKG_ifpnT__4hmULvXMKPuSkA0gcjgz0GSj2mVJBYH-JAwYharouo6M7QXNTg/s1600/vid-0547.png" /></a><a href="https://youtu.be/dF7-6zlsiqM"> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjkhtHeAeW0-XcvHAnMhKRgC6USEYLzZ0o66zd470MZF7EMsCemtlpGzO2ea0iBU30smrnkx74JxAdTlFOtNjuEW0G85rut1aWAxqbGXFxaPkF-3Gt0FcUPmxASN81fJdwTrsJgWAszOao/s1600/vid-0548.png" /></a><a href="https://youtu.be/orKv4SvOD7A"> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiiFu1qv60yDEUBM0vNSl3Bx_gL725z0bewRB3LOIVIpBejkzsHRqC87PUQfl32uKvCBFB94JC7M_DSBV2CYXJWoNgVu7j7nG1uETszDW9iPpQy7pgxG-xVWwP0r5Lw6IQI-a8Dd2ICIDs/s1600/vid-0549.png" /></a><a href="https://youtu.be/ybwXhYprcmk"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhc9g6oeYW44HOmHe1aLp1cDM5qcrcpoEqzvJ33XhUuoAw8OY960E-x5ivgQcZTB0zQFkIbT7GdxrOxQa8xW8NxYjUnhcxNV39Cr0fAvLtprJ_onp6qY-NK_DVkLFqHxi_cVivy23JeIS8/s1600/vid-0550.png" /></a></div>
<br />
<br />
6.1 <a href="https://www.openalgebra.com/2012/11/introduction-to-factoring-and-factor-by.html">Introduction to Factoring and Factor by Grouping</a><br />
<br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/IRaQYyrgXDw"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhLlsSm4wFuizJpUE8jbbRtgugztUFBwmmIyx-Y8AlCMr4YhNqHLS-EpeEl5hObIDQ-dJ_t4-Fq-QdrXN8OCRNtQv_pReqPrI6GAdUdlfdjg2xiKBCgYlVO35W438L0qS8jdXa4bil9zP8/s1600/vid-0601.png" /></a> <a href="https://youtu.be/gEcAhvtEQok"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi7QVqBAXn5TBTILMdEOBKUd3e62WkumXkkf90zXelu8zbNDwVWMgVIVPHmvj2-kKbUwwf8rKgkxk4grAFEjYFQF4eydKTuze_ey_4tua65f7Ev7SDWLX4H99WOCM1azknmFhVNAnIRjAs/s1600/vid-0602.png" /></a> <a href="https://youtu.be/p0aD22tfqZA"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiSACoX25MFFtl9h8RrCozvNLMBONLpU2Rr8gAwJpR83POxmXPYiHs-ElPriAC_hWdaUGphimukXuHiev5oABwcp6-nhYJnMMYIGyWqwW1T9qHv8fTvgCKl4gi29w7o7liIG4o3ub7Exls/s1600/vid-0603.png" /></a> <a href="https://youtu.be/R06K0FLNo3o"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjtXDnRbZ1e-YE1lHtxlV9ugprIx0eLNfSTntUl8Ez3slLCE90IJymLobxyqJex5F0AQwJth6a5_naloEMXftwPOB-bm-EYpCtLrm1uYRvahh9ne6EhAtRecmALP3wcRQcOIDM3fkc0jC8/s1600/vid-0604.png" /></a> <a href="https://youtu.be/zSzoZeatuis"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgGKfRwRXR-4_49MSBphGDMiwnDeGZAmsUNb67I2PSYWELAA5xKkxS8IRw0QrCTaeHozIIJtDeRZjm0hLg03uW31LbVxJehL7csnjKpUC83WHByXH1o3LmVLvIMnQdHpGhLYUvgi8n9hw0/s1600/vid-0605.png" /></a> <a href="https://youtu.be/ruNZeaEyzYE"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhlDwBhKAsDDvacMadaclOYGbV7uFlpB9MyekGS6fmoyXkkTYdVJgVMDxIom1zzyRGAGPBlAbUYwyDU8iPdfy7dtUCKBxm2zmxk_KcctGAq7bOzad05x_CZMggAg_g0arwtxrcfGF6DKXQ/s1600/vid-0606.png" /></a> <a href="https://youtu.be/vfUbbh6VLOc"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjTp6QOlRfDHtfzu7gJdWd1147VeZ3nSFWddFo_nxhcUuAEanSmY4QRNZ4ElORUlivrvTQ7TbcX7ZG655WIJv4LGeb_twi2HcjjS9-PxL_KYOPWj5U-0-k1j5am17FhDx1f3xp6sfYEi14/s1600/vid-0607.png" /></a> <a href="https://youtu.be/-cZJ7Swm7js"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEghjKiG2PcWmNSUHunFz7dYpw2mfFfZ50SKIBTlVHmvxPB7crHrmWEk3NoSWquvfoqXMPjSOD9cYeodC-sx3cztUeDJx-plkxMuuXsOnA3b0SThDj5snPtlHwhG_BDGQ8SWQO3IzAzqB2U/s1600/vid-0608.png" /></a> <a href="https://youtu.be/g6sjWuqbc_8"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhjEgmbWK7PumR8Xk_CyYxStMz4rOS3YXCmO1wd9XItR2lkVbPN1FnKOjyZhKTPrHtsMNprnFX5sNGsUskTj5jSXlqesPd0B25i0B9fz9RNlUbi7Bz9OIW0SLFJhfift6qd42Nhs3cbqGE/s1600/vid-0609.png" /></a> <a href="https://youtu.be/j_jF0vkSG9w"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEirJ8JKTMwThv9orldNAxCxak5LHsxTI_0JDbplOCAfxxRc6XmkK8xM_ff0z7z3Ajqai5NNW7j3r0iluaVjH7OvHsl715H0FhumGtMRfcnnbc6qqyrCtqLodWBX_M3m1fUyodsHdL908ig/s1600/vid-0610.png" /></a></div>
<br />
<br />
6.2 <a href="https://www.openalgebra.com/2012/11/factoring-trinomials-of-form-x2-bx-c.html">Factoring Trinomials</a><br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/zzXOYqbRja0"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEim23H3aKjr2MVZTE3sQ5yKnOCa7RsZ0y9Tfsp-26X7i0-gZNx3E06PTxHyv1r6zadphc6t3oNzl4Zf3qsyXtQkFXgHlfQLLtmOu8tjo0uco_tHzr8IIlRxxYF5CTmuVXiwi4ibaDpyPS4/s1600/vid-0611.png" /></a> <a href="https://youtu.be/uJGbD-wL-Zw"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQFfZEbmbvIS0jQhaDF_B4Fv8wEXf6mZDIDSSdHiOCy-fH4sElEAuZfelfvMSwG8a2eakhiZ4_Ugkb1iheGnlS67rmEZ9GdlK1YSMz-LPSD4HDhoitPRNtQ6qQtLbwJPVCW1Sx_HYaKAE/s1600/vid-0612.png" /></a> <a href="https://youtu.be/YM3LjzXqFL4"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiP0B86s0HR9uNjnSLomMBIdOBz25ABNKicir7AEQhd5NBNwsiVbUWAR7MBf7LAHFd_Dp-UVDEHrtoYm7jyEmjAhs-AbC-boNfcqhWcvlo4rPZpPTT0SOM9CDwh_3WEnui6RtRU8arMqns/s1600/vid-0613.png" /></a> <a href="https://youtu.be/QhW-CPOw4-4"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgmM-xOP9_Cf5tciGcvJE3Rxkbsqv5l5KMTr5z2Bo2ypsez1ZDdNoX7wBXob7lXblliWPkaORol3HSduJ8ptbHah7bkDdazvBk9mEIa-QO2tWFrHnsQIX7OkCS5zFlnxEctRV9-Wk_jeWo/s1600/vid-0616.png" /></a><a href="https://youtu.be/dXLwNRhjFM4" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhb8KGX4dHyVSysI4JtONZpr-DFn-kbHsqr_A8OsgNw9G4o4wXjtV0nj4L2RrelNn5rXHmsgY2mTVL5tE1CxcqBy7WLXSn-SXgXEGvgzX-M_XHqH3DWXJHiKXYFU88pIJKK8zq73fQIEpM/s1600/trinomialFix.png" /></a><br />
<a href="https://youtu.be/KTnmwMn2hqk"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjIsXHfCAxgjQsnh-WMs4U88aESSKKoApbhKS-7MtWVQ032RK3PDJ9FMNBnedIF84CdcSvRoFO8f0JEBwlCcIJ-_jvUm-L5x7kev3MG5rueXpwHSgVLH00L5b_FEb603kKCQNjZIdWsW24/s1600/vid-0615.png" /></a></div>
<br />
<br />
6.3 <a href="https://www.openalgebra.com/2012/11/factoring-trinomials-of-form-ax2-bx-c.html">Factoring Trinomials ax^2 + bx + c</a><br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/LFYTG6PC6Ng"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhs_I_DDYPVRfvNuSATAv7SQWSe3CWf8iF7tZe7bNa0S0OOTcq-EMrURRr7KaKJphZ2OrLS7ZXnEhLlN4wuB3rYdCHy1akI3CXfgwOIYqDTAlAjwMDmEyVGcDbEsZn2x2Z4tz5dTwvRXyI/s1600/vid-0617.png" /></a> <a href="https://youtu.be/w01HSb5kSio"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgC3l5YOa-GlNQG5F7gY1zT3I8oPD7zCN_c2J8cz7b8aebm9gNwvhEI8gLhpjm2UY8l0Wada0g3wj7UH1L4uHo1BBEkV59WqPegIQbulASqHKJaSuQB7shz_uiEcDe33NlQ5MPkrikqGoM/s1600/vid-0618.png" /></a> <a href="https://youtu.be/kjXWPpfCxWo"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXF-u2aumAMdahGOP-3o1hBtUrJvKiTG7UpdVA5CYdUanHnrATRfPcK7lr3s9MFzEMpJaglC13vzYgAD5fuLRWzFYP2LH_mj5dR5RsHvSH3UTWCKt8oedNTLIUjaVR5sZFNJse3PuU_ZI/s1600/vid-0619.png" /></a><a href="https://youtu.be/MFmYtkjaaqA"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhw1Tr14YquryuU7TAqmMBtalYi-4EZ8v2vJUwZ3h3Vl_JQnyQgxY4SjURkHntkLhm1xOKOdB1Ss-BXo-zU0PrKaPvLv6M3vQ0_3_M2gK6nSzGdVvXnJGOhaF_hGxXYojaR4yqDkCV-ikk/s1600/vid-0620.png" /></a></div>
<br />
<br />
6.4 <a href="https://www.openalgebra.com/2012/11/factoring-special-binomials.html">Factoring Binomials</a><br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/1d-YAFZrYYU"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhs7nv6UKKjJ-3Ceh-Jxgj41qjKiLwP9mXyVFO1I55b3omDWfYobDGXrcJc3A7JTP7TcVsWCCHAWP3pTx6VcH-Cfe9nGw3WtRpkaCOd9mexV8ICN_9Rn6zjSd1AuFDHWMCkPuCz5xFP-Bs/s1600/vid-0621.png" /></a><a href="https://youtu.be/dwQ_NZfdutw"> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhOKIaDietCUkk_6RFn_hZMxpynf9PtqNebxZUGmdW2btwSK4jkVsuXrPrvYN8EZzwugYoDt8AzXJRY4utWT_D9hfn7AIL6A4XRqYn7MXYjBlVRKB0-U_ILlzrIalpcuYkIxCxPSpwpyzs/s1600/vid-0622.png" /></a><a href="https://youtu.be/8bMdida8gwA"> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiAnqU46HhdGhWx2SaMlvBr_e94zrWd7cIIC0Mo_VcIYnj3-up1j8lISxYwX5SOrLc1MYLdugOjHbD-49vuCRafCDe1WJHSmUR4b1l3veDdVbu3-oBlM9ZjFKH38Gpmm6jW6dfmkW-JWDo/s1600/vid-0623.png" /></a><a href="https://youtu.be/H1mIlYlF1vU"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgxltQqdzQmU4FeMIZ4cq1iYWS4qPqcPxRtabzfsEbXaMfsEPRAHQyhDBAAjRC8CZnLdtZcTGJusgbpphXlBsSe0Aj8hbvlpekJNWUrfzDSfAnrMPqrHUZKm_-k_D4cOFr6HZ_phPZtxOM/s1600/vid-0624.png" /></a><a href="https://youtu.be/af9zJat-XGc"> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjnMbpo1WT_5H0yUdW5IsL2fX-_LEnOKwQQU9ZAdt7XM9jx0mIM-zjuc0x4V_IRgvDNrR51XKeIO9Ly9SZXfS9_p0QnFr82E6fzZS9FcR22zgRV3PVVVe4oLdrZGke0qTT__4EhaiIwMyE/s1600/vid-0625.png" /></a><a href="https://youtu.be/DX-NUBzLwII"> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhTbKcuX_5QURxgkxFPR3hZ_DMY67iOGus4j06y-RB7OcDLET5NNWyMazQWgJzrurSiNAn4jMgMHVMC0xbbJYRB1Eb-OKqOHEEWeZqGmElgsnIB1DQVxllDsZ1ioE_Ia3Zalihp9mt3liQ/s1600/vid-0626.png" /></a><a href="https://youtu.be/cLn_ibYPwXs"> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPAgbtOqRc3U1mYMpsBR7n2VMxL-tfrEi6dZBMiqg85WGa_MgEzTP2KrklT57-0VKMJ_c6Yg0buNdEg1pHvj6vByD6P9whORUzz6cE7EbZWDmYT-He1Sxwx1tqaZyJ0jWQVpSnB_QcmoY/s1600/vid-0627.png" /></a><a href="https://youtu.be/3-YvBxbRAlo"> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhlo6cBrj5wkAj1PjYwiySvbJ-oJsR5ZZIy7LuluThkRb8pLeIdU6aCVZIABTTFaQA5UjP9TsGlDAIOIr8eheruBz2JEvaEvjppMon_kpkI_IhVYcD3znlSfJzRLvMO0kgytVnDTlrGPmE/s1600/vid-0628.png" /></a></div>
<br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/SPmPfQSdF9w"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiAOeyFsaZbkZDYj7tnpN67126U3YzDWGhGqx9k5t0rV3pt5OQmLm-BKS8OKKubQFX781MvEcNjwVn9_yOy4hC5UKw9GkyC7kAaZCsW19zc5TdcsGqjtblJI3U2fkobBxaDD6DV355VjOI/s1600/vid-0629.png" /></a><a href="https://youtu.be/Dh5PJtGV25s"> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEizQQnURLPiJ8ct-HykPZdwMkrwotGgrLJo5lg-qdxKPyL_sOkSVMkwEUReS_FJ3J7IINMIp5CHWJQPbMB21_tsqrjolSQzbrEB3TagiAm_GZf2IZ21Ev6eC6OD7hfInwoIYzXl7VKQGtI/s1600/vid-0630.png" /></a></div>
<br />
6.5 <a href="https://www.openalgebra.com/2012/11/general-guidelines-for-factoring.html">General Guidelines for Factoring Polynomials</a><br />
<br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/bnKofkjdkoo"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjvK-HDUlX8H8CYEjJ5DXAQDJQVTveo4mriJLgeKuCB44JJuIvqP4x_LfYu5DaPwUttvcqAy5pOA7GE_n1Mk_BvKOyaO8YsqAOEgSzt5NEN_iVJxUv2bncpf7e0uzGZ9ER2P6QmxNT_mNo/s1600/vid-0631.png" /></a> <a href="https://youtu.be/O_OfA5roXw8"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_eZ3NQ63bhuTILey01ybbOScL10bU1FU5Ss36-65NJFe9zyvyyUlflglhVKq-iYgXDBT8gps-kSA-a3HDNaqQFYEzrmMhNxNLgFgH4ePHc9YtJjKW3Jo8y7x0uVADxTDmxYsiFm-rEQE/s1600/vid-0632.png" /></a> <a href="https://youtu.be/i9wkTfs5VkQ"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh3-kFx939hP5H44N_JHz1xAvhPvQ99SwTxujDnrE_KdFdltokG-BF-EYh0sGiz7PdsgOYY0ESn1gUP8hy6Fw0n7klREKzpdqVdEalbxno_mmo_pGSQ0FDovNClebAqtqab1bdsssw6U9U/s1600/vid-0633.png" /></a> <a href="https://youtu.be/uWjgZRX8fko"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjHz88YXkUzvsV6ZcLPVsxP1uO4nw4ByUQM5CuvSeLKXttuEOEVE5DrgSU-7QXCFHhlP28H2ByhuFMCIYcgIVq0564Ymf6y1X3pC35G-Dg6_IErBOy8_n_kV7XWELjFgK-rhAxWcWJd8yQ/s1600/vid-0634.png" /></a> <a href="https://youtu.be/duWegCpwkzE"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgVnH5Yl5MuAWwJGX6PbdoODPbi1ewX5LAKOHCy7daHdRRWh3VPzt46_6e2iw9bmMWBwq1lb-3PKsjMxaIsRfR6VasJmiXKLtSWvzAQgVqkm-rNmkcgYWnojWNQVIVH71UEfbHg47wto0E/s1600/vid-0635.png" /></a> <a href="https://youtu.be/k74gQoF78lo"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhsBBHH2v3WnwiVrdx6hj63iA-sHbKWVZB8SOpB8-qO45i_bHdTACT1soL7KfoqJeCu4hojrg5zDk9n56pKBdCDr41P-22ylFf9RJjbbLzCzr4zBA7aeBbW0-hPGNycGGHj1tuGgBLdM7U/s1600/vid-0636.png" /></a> <a href="https://youtu.be/SfhdPseviEc"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjJmiKbUupqokc_ztlzTCGnAtWp8YDOPuoyWy6pkrU-vD0AgYxPRq_9b_7p4NpDEfufD6IwkQyDnerFnsQ7TNXyIFjcOLUSZRUcHqPQByf8wE6tWal2s4JkS6VIjEU35vEHOtRCqS7Io3Y/s1600/vid-0637.png" /></a> <a href="https://youtu.be/CoArgH44JGE"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgYnpK2mMOTlFeXzD7_SaCz4vJ9jiQ4bfAWOZkiy_zv9JO1fYJKCaduzYeWYkEPr6JFQj_ucT3R-S2Y2mISg-tno5dKiV3oHG5wnYTJlQn41FvWj3mmO58ICzmEJJwtesLI3ADghoXGy9E/s1600/vid-0638.png" /></a></div>
<br />
6.6 <a href="https://www.openalgebra.com/2012/11/solving-equations-by-factoring.html">Solving Equations by Factoring</a><br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/8fdZG74QDds"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgUyVPEAhySccZGz-0D9HIWUZcqpTXfAvr0SLp0h4909bi_bVnJzb_1DxC6jdpQ8uW3DbJSH3v4reDNYQxmr18bT1d2RQrcxY8EZfzY0h2sIjn8rw4-efMMUgBR057_48JDBAQCO8Wksyw/s1600/vid-0639.png" /></a> <a href="https://youtu.be/99XtP638Hp0"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhakgT0WtR6tYbB01y4VcKPYj1P1d2wLFxsZSFbk_Pih0lP7MoIMB4AmotUYPV8P9Qne8jYpWNJ7hjKKGjh7Mm25Bc-k8R-Nry-E0o20LduuhqGmsrvlBcx_uGsJ3VsRfwRnf7zWnsRtEo/s1600/vid-0640.png" /></a> <a href="https://youtu.be/WgHhIvotyuU"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjXAitX2K7XTyfjRPIOac2432Y2-0XPOHHofEwOkZTrodKLBoFKDzbUgoZdBYZrqX6EvLjc8OtZ2ZQgJ0x4BouyO6ShiZD5TAH8c2YtDHa99D1zMGwpmRlyDD4d6oyXJMehNLPySqLpZ14/s1600/vid-0641.png" /></a> <a href="https://youtu.be/6iKbpbf-Z-Q"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvHnrCAuQbfhsJRp6g8MWM8EgBQ4gAIlSq_xRyEVIT0unMbGpgNhSf3vvLcAl8aQqJJaU_5pyBxnt1zF76plJUhxDz9GleU3mS44HmG_b3Uj3Ln0QYDnOulAkmS-ssaaz9QVKhFvdhRRc/s1600/vid-0642.png" /></a> <a href="https://youtu.be/Sm5kUMuXet0"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjPIpZbmdr6bHN9YCVt_FN4tO_XrNbK8-SFwmWcv2gk0bVBRWZrxEuLGocpvAaD0jyca48MDLEk4Gp57BEUrAa98pepEPDvHZwijWm16y9aEYXpol-neoPW7g3vSR3Q4LIZqzuP3E-AOeQ/s1600/vid-0643.png" /></a> <a href="https://youtu.be/6mcKK1HYhlA"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiiAeTFePhkx9NycOJb21c0CZH0rm2UPEQlK8mlNdThOKsG7mY4CSNBsgUBFOmFFDBrE5-7hVm0gUZP_SlL2C5bW8mmIrQeV1Ys2QykIA-s3CghlYdy46DpgNDepJ6yzJkmCQsEZnUbm3g/s1600/vid-0644.png" /></a> <a href="https://youtu.be/rq0KOrKTOzU"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6UV8PuOjTNheA2HAjTZ46Mwam5Y9xTll8xrqYTVHOA7mEhxrTdcybVYkyIiiA7tga3viz08yLr1a4eWHEXyPnVaFni6d3xcee9ph1H-vSQLT9lufPI5m-RMmjxC6FwWMWG4Eh0uTZEjQ/s1600/vid-0645.png" /></a> <a href="https://youtu.be/cjubGBDjtDY"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxGayhAQyUM96HVwjK9GHWuCjN64msPIHfNiA8ITMTjb50mym7TuOu410PbtUNQ6H-ENlAidNL4NihuL_yMry-PR9NsVaQHZyUeyZvUjFE-_exl4PXDs2vE7AFVaC1nemfgfPmhX3etck/s1600/vid-0646.png" /></a> <a href="https://youtu.be/QSTjG2JLloY"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEircDF2WFFOsIqYPYAk9CbDrF46WBwy8rIkPNtKnTlPWpoq3jTvckK7XL5WC-NO3wRCSU8M0bp0eNBQQnZJg7Eb5S03Tp-UDS9oPWp9dLmpTVGLt6-_pSuvYRTuNs6E0T-g9qtyz3EpSnU/s1600/vid-0647.png" /></a> <a href="https://youtu.be/m_rE_HNBr4g"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEja20eB5dmpoyBc0cG4Y8Boib20k-iJcT7E7KkeXmDTQ1AiZXNrYFU7ASB_N-2nEawnz5FUL82s2O64kQ2cFibkPyvsvAG0m82C497dsyjNPWJzoyHFARS9rsV9gZH_GTN_PZQJeyEKGzE/s1600/vid-0648.png" /></a></div>
<br />
6.7 <a href="https://www.openalgebra.com/2012/11/applications-involving-quadratic.html">Applications involving Quadratic Equations</a><br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/AFVdqy5mvY0"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg3m_0NOgaiMIDACGgv-EyaVkE28gLLWVF_iR-al67IFr-Of-E2_070tMKqWpI_gcpchYfT5cuHxMDgWz-51_C1x-aoEw9eG1zfeW20weM-YWh1SCgSgLaUCgpPJMqrbLY_pfb_vhKoFLM/s1600/vid-0649.png" /></a> <a href="https://youtu.be/1Ved8b2VkqI"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgS4HpL5acAQ08sPh8Yr7BxCkft8BFUa5sOe-zQ3a9Amp1HktozS-c1g1UgXTksJo4OZsfepQQSwLlC9_hcf0rwot8lRZdEW93pJS3wdpHmZHS7UFGw6zkEKcZEkWXC4XYbeVBPaYeEEcA/s1600/vid-0650.png" /></a> <a href="https://youtu.be/rwohMXc13gk"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-aJXc0jIN7OOQ3KKbMOei_8JHLx-iCZ-CKgDxtHh8wzoZcyvjB4fGxiOEhW0jVsN0Ko0gfEKq7-_uqiP60TchtwysmJIXclqXHLoRsxOwSgVThS337PBQDTTqzssAY5cWlmnx4-ocIdQ/s1600/vid-0651.png" /></a> <a href="https://youtu.be/DDZcpCfxPVc"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgKZ0EHxfXRZW-BN_3vPt5hZpbVajA88QEN_KbATVluaxLhDAs0Q80rwv7F_NMYGF6cg2Qgc-IQNcLFxkXmPT3AiZ6FLgkqWatfOlzf9iDU-KQycPOEsIlzNZePsRtnvkrkgIUjHJA4Wow/s1600/vid-0652.png" /></a> <a href="https://youtu.be/sepyKpPbF_4"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMRcoXgE5WAHTUqPiXN-OjsDNKXM1cW1vLC5DBnIw0LHagLgSwmZS16jKZqUHSjyMtp-aZpW3kcNm4-1wG_N8qlPVYqA8qJWFnXzSrJsooi-8ibmnRAxZ5o6ZIDCg7cQeH_8zZ6PEbXlM/s1600/vid-0653.png" /></a></div>
<br />
7.1 <a href="https://www.openalgebra.com/2012/11/simplifying-rational-expressions.html">Simplifying Rational Expressions</a><br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/S29qWSUmpTo"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEglwRptTxY9DIZd-GM9CSJ3t1_rO7lzPhFS1qBu15mHeLRdA-BFw0PLlIdSUavC1ACRrLka1_BTOX1JHZWfOl2OcUOHFISZ669m9kt-nrequKBMqHZpRlcWCIuGn1TXyUdL2rWMCFqJ0GHg/s1600/vid-0701.png" /></a> <a href="https://youtu.be/L9uYsA5Dqv0"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjiaxZQwLLIibNzhib-ev5K0JL3IHWPPI9hihS6SdtnSZDZImW_zIjrLyC0DVKMHLte-deXuilwJTrxUWKyGLzqZEHgTyU8U-DYQiLhvqXeBzvkMnfYDyAi90XJXnjdPCjxIoRd6sXcjtGQ/s1600/vid-0702.png" /></a></div>
7.2 <a href="https://www.openalgebra.com/2012/11/multiplying-and-dividing-rational.html">Multiplying and Dividing Rational Expressions</a><br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/_VcgLGF9H4o"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhYD-2kAW0bffBbvgHKYM29sxc3kOG-UR2QDtBsPZYMPc1ISJ6x1ZJkcXDwRzEe-T7QeGtUrilwqVBq20e3YnNH1yavt-1qFxRsyA5Ru6ee46g0AVmuMDZCZpWGl250Pg7QYXVDiYNugg73/s1600/vid-0703.png" /></a> <a href="https://youtu.be/0vD0Bp-t3mY"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgMB-fJjGyUf3fNF_sBSf8ic_cNJt_1EjU1AODMSUp6jJwp3PhJp_328upDWOoORvCx8IGJQlzxA5ZEiYk2Woz8r7NBjf3HRh2xjt4wEYmSlzEk-JfUpKP1O8bNrSGtklEahIOD3-IVhOrl/s1600/vid-0704.png" /></a></div>
7.3 <a href="https://www.openalgebra.com/2012/11/adding-and-subtracting-rational.html">Adding and Subtracting Rational Expressions</a><br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/S0U1Io-FyRI"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhp6p40Zoa9HH6XDmunx6rC5bSwp6jjZ2MfvW1FOtgcI1WryLTkDFVB-8JJoVnXKX4aceIKi3X1z0FHMm6N2S3Ofuwy9G9KiMhsdh2Ry4nu9-RTTm1u6fr1loUoL_dm2z52ZpfQOi7agO9M/s1600/vid-0705.png" /></a> <a href="https://youtu.be/Wq-StmIyHZ4"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjtwLsBghO0T9zmR1Qt6BWAWcB6vQqjEPgcT9zii40OVd9S81f8jr36-nz4ewfJaBHkznD3x7tPzPTqQl3KQ5Aaltl4Pr-q-eoD4J48H88uVpIqclkEbidwGG3TbbaKqyYk17ZU2YbeeW4y/s1600/vid-0706.png" /></a></div>
7.4 <a href="https://www.openalgebra.com/2012/11/complex-rational-expressions.html">Complex Fractions</a><br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/j-nxOYJVj_c"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiYO71yNXXBktt1aZK1oBkA15gDPbQgoEbjnnkeNgpuuQnngbNpi4ixqGzwAHmnDlWcIyFWNRWvw7D5sjoSzxIUBBCPDYMVnjycBuBsDFZxDialBkNxjJikADlxjMR0QaGEmos_AZQ2hTRj/s1600/vid-0707.png" /></a> <a href="https://youtu.be/p2PHXbtpu8M"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhqoBxU8xLVTjhItLzUp2_UK9SVx6UVs7gG6eawY2H6hSMSo8ckYjMJOSOZzIQkYerB-SEJykVuIsv4alq_lSen4iUZj5i1Y9LWwnp9MA2Co-IYHhdKUdYzB0l83LzyeQAh6sZIfTj4x-se/s1600/vid-0708.png" /></a></div>
7.5 <a href="https://www.openalgebra.com/2012/11/solving-rational-equations.html">Solving Rational Equations</a><br />
<br />
<div style="text-align: center;">
<a href="https://youtu.be/Hwm2a39pRMA"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhpUxtyro_44Z0_VQz31sXs0klEqizWIjTc7JUJ8u_oovP87Bk6R56MCoyN6Cpg2oQbp6GAihwg-irjVhkUXAe5kfA-Y8sqVNckDt20p27achsTE7K-56ekO63lK3b-YxHs1pzPMZveKKFh/s1600/vid-0709.png" /></a> <a href="https://youtu.be/xiVZK3vZe2A"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhgZ-O1V_J3CcA_KzMf85BlxN8Fh870SrJqc34iSIkpr5iB5BtdP2ARfBbI_0wZD_Ydqy0G2VfCFl9vaHrkXgqtrW94kjpHNgy_dd-MUL0ohlUFH6_5GMSr-K-FM2lp6qTLDsMAdp_ETXui/s1600/vid-0710.png" /></a> <a href="https://youtu.be/oLnodCwNe54"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQYfPdpaPJGruU6vb89bsxl1f8hnpmB3svHcVxfYXhQFe7O9CWtQ6V1n3gr0ura5HESJDVb9566TrOomdZ1h7Qg_P8NrgjAIsPd_5TrKseuPzjFHcc-vucL683gziAYiHCkEIIiejIF_Wb/s1600/vid-0711.png" /></a></div>
7.6 <a href="https://www.openalgebra.com/2012/11/applications-of-rational-equations.html">Applications of Rational Equations</a><br />
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<div style="text-align: center;">
<a href="https://youtu.be/b3Gj6UCrTPM"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEglLPS88LIwDoHcuHiPRWVBQ8Lq5qQNG900_mpc41Gdmc-lnIa5HV5XEdXVJ50H8Xkp7jdKxTQSVR0mBQYzXVM-Vksc9PywsVPpjZQ1zuLbN1Y415U-dHO2b11occ-BtY1bRZi-lUBJnF-S/s1600/vid-0712.png" /></a> <a href="https://youtu.be/77ICZshIosI"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi3PS1WNrRpB6BVrrLTtVEdJYCH0kwxQa2V0k94qyyvqLESZYwUFUxpZ3ujPs1J1l41Hda322KodQE538b4_4AH7fBJfEhKk3fpITyuQXZrAwY4TQxItXvzVqtRRIicosLfB5Uuk4QH_94S/s1600/vid-0713.png" /></a> <a href="https://youtu.be/SqA0hltJbT4"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgFtk9x3qrJzn8wYx0uzAsQp1fJGyLPs52ie_N49UNW5e7S8u4htQNIFbaM0Ve-uAvfPysMzE8yh1caJjmyGqfzmflCY-ybT8WfR9l175lUTZnxtG58FU16LT4f5T_tJrFjFiVMfk8LF0gI/s1600/vid-0714.png" /></a></div>
7.7 <a href="https://www.openalgebra.com/2012/11/variation.html">Variation</a><br />
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<a href="https://youtu.be/xVNvC9axFwY"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-_yn2QDP5E5dXuwozTiGyq9hHQh6LQIyYM3VLHmts0ieiv8SoY8Gj3V8WMZ92_OGQTyxgAnMCIp4oXvpVEP7wqOMRlwKCq53MC88bD5IWOTW9DeRM_U4S1C02NUd1m2JHAcLYdYzGY93m/s1600/vid-0715.png" /></a></div>
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1 <a href="https://www.openalgebra.com/2012/11/radicals.html">Radicals</a><br />
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<div style="text-align: center;">
<a href="https://youtu.be/zGPH82DZjc4"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEidKsokMhhRsKpoCWMOD8PK96708U4caG9_R2Sz3LdHdGe2-VafCucBRdM6b5l1BLIQNtCTyI7kYtlh0p2cArtrDjBv4I3Co7zIzi87bn2L02gSrzjUpMwtzyJ_eUvLYwGeA0YZPKmu0iQ/s1600/vid-0801.png" /></a> <a href="https://youtu.be/JQrojcW5Hdc"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRUSLlJgqJLVVSsDsMjswlycMVPDtfrZkKMsbHv0ylDFog1J1SNg8bEA9vJAbI7hE6pAP-Bj2umHQC7z0WdaBS1woLoXXJsmBGtFbJCIvvQwoAKWMFlb0ypB6bY_XO_smapijG5rzD-qY/s1600/vid-0802.png" /></a></div>
2 <a href="https://www.openalgebra.com/2012/11/simplifying-radical-expressions.html">Simplifying Radical Expressions</a><br />
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<div style="text-align: center;">
<a href="https://youtu.be/on4bStap6fE"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgNhBK8reD3ha9vD38aRFwI0MJ3_gpYavNx8I3biDiXg0Uvh1ZsDnLvXWbkjD247KMkMhMsWBJMYUR0EQt2QGeu7RkV0Bc1zZJAKGy20AulXnH7ArP9uZWVay77QJwrt509NK9b71LH-fI/s1600/vid-0803.png" /></a> <a href="https://youtu.be/g7df7MQDsi8"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh0x7eNE_GvKHhj9MDSLDa70DvZWPgWlV8Xt6y5K2waF3AkpHDvv5NGNM-LdSN-5MA0s5GhqRwKB3v-71mhdD6b5BBrtAjOhSP6BxLSQCG0979EjS08wGkcd3Mkk2NtBtLAl2WdjUiaNPw/s1600/vid-0804.png" /></a></div>
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<div style="text-align: center;">
<a href="https://youtu.be/gqSrCOA7BGY"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtZSYkz9Ty7HhPZOo1BrFxX7iJSXt_ILkee2NfzylhyAVXxKaz55LlA_06VQNWUxlwAmZn_TP09_Nd0MnNB0hC_LnM0aY-IVdLcG0BwjEzKwz4chrN82DWOYGCPF_wCVYDd6qU1EV-qvY/s1600/vid-0805.png" /></a> <a href="https://youtu.be/iVT3UvO7X2g"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiUDmayQUviBxq80mDEhlGXjDOyJ1CWemqqddsmelDi-kF2zJ7MakVfVeNX4vqD4b6mdyfGkTH5NO1h2H94gfnzkQf3G1rv38z_yLvGUr40YhTHuTwQF4zzoNY1s9BKZPiqBL5BTjZkKEs/s1600/vid-0806.png" /></a></div>
3 <a href="https://www.openalgebra.com/2012/11/adding-and-subtracting-radical.html">Adding and Subtracting Radical Expressions</a><br />
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<div style="text-align: center;">
<a href="https://youtu.be/wQDMPoAtVCA"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjK4NxTruWEoXvYBOfS5HrsnMaZ-cYYWZP20S3m_T1VQ8uC0hqBX_hatWymH9t3EV3386CcfT3fr5faJQrRAwWDsE0yzCpJe_oKpWu5VMf-M0LEKbnMfcV0WST5FtqfQBYJT4L4-qInYgg/s1600/vid-0807.png" /></a> <a href="https://youtu.be/f-58AV9neSs"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjwhN9OU_wGGJBog-V9XcVgvr6PEEjX3Ukzrb5nnjANrzcygQQQ-vNmies0-L1YmgxmM3ZJHt2dOXxrAV6AaU26Dab7sTXA-AmVFAh5e4L_VvBBSq28cIAYYIAn2j6li36uJUgHixOIISQ/s1600/vid-0808.png" /></a> <a href="https://youtu.be/qeaXrcybGdk"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhDjQt9vdoduEHwVfZtMJr9ZKyCQ7Tuzu_cGSTXa_nwmVShqIduacEfS9JeEJ4-Pu19VB_XShxlIp5QRoY9X_VoBaZFb064yu9gs6kLSmUlxe0JPKKd7j3TeY37V2XuiMBxQ95uxKvVmXg/s1600/vid-0809.png" /></a></div>
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<div style="text-align: center;">
<a href="https://youtu.be/Al7V4AkErdQ"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjlSN9-atix5bWOZkh9Vr5RIGihCBHFLCqlx4LG0kcSREE7P6LXhpcnRHfzpNY3mRfBzWVo3uT00_F0ebmwdDmSo_Xi0hsDavcRYfrPH-gDJaWkMqjkIe7vxKM5_6miK37OD83d-Ao1X8w/s1600/vid-0810.png" /></a> <a href="https://youtu.be/Q02WlLtm8fw"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgBIarjMigzfRZXLvdZRtAwAFt-s3sggDXzF2p2d5dN98cdSa2wwNUXVPSuWVL1AprXn4_A3FKM_lEo6zmEtPjz5f0INmPD60UF-xOo04gSs0Atc9MLBB-E1oJP-LrJuAW1asuYg4atrCw/s1600/vid-0811.png" /></a> <a href="https://youtu.be/PxjMl2pROLY"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxwM0tOMaK1eva-IdPqbXjvlAxggyjqVsncmF8j8a7cm51SGo8O7v3BelkpAoidP2o4rbYWq4_X-dqiLT4QcEhLL7aHVMw1uWYAf94yX94_ulMXUaF13SwK3Xxis4qKlYlE65bXqWHP5Y/s1600/vid-0812.png" /></a></div>
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<div style="text-align: center;">
<a href="https://youtu.be/lC0ZOWzTm0Y"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgB3ska9LnqFlXh31cZvlwQTOrSMT-kaDhS22QHSm42_SJvYJeYwQiw7e745afe32B4dkTmdL5WxTyNYCkir-djoQbGYaFZKASZXn0hcpPJWniHMDmF8HHm2b65EA5L0qE_CcYq9zim46c/s1600/vid-0813.png" /></a> <a href="https://youtu.be/O2z1ro8JZww"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-VLf-5Iur1290_ZdjKhezCECVq3d7ac-kIvKyKjBATHO6OxBvALniEcEUupfVEhTqsM8SV8k6t-4OjAiS5zCuqyqaVtUInvA2wEnMYFi4hUmNFLYeDGZ7JU9-vK_9KKQTNVCym2Ot9aY/s1600/vid-0814.png" /></a></div>
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<div style="text-align: center;">
<a href="https://youtu.be/3a7g5SwgVJw"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_5dnpKdkXXEKC3I88K_aGPcxH017yQ-7QRfQE0iwy25WJkHerG7wMj7HMZ4vln9affgrHJlnKgNa2oNd8nofF6UMbblg-J4GPOoonYAp52xQSo50yC8NBOH3FcRDBm4wbmsn0Kuufyvs/s1600/vid-0815.png" /></a> <a href="https://youtu.be/M4hh-5mvSx0"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhTY-0c-kjKK0ed2vI0Y4OUUw-0xWcg4o8VBGnOaCcvv72mTWw3KtaMlSsJUhsY0my4SVtHqO24szx-WLZIhoCkNWGO275JwFWY4qSZ0pFhyEkzA6wpkZPsSGGF_uzyMoQ7Ih_WFlrW3xQ/s1600/vid-0816.png" /></a></div>
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4 <a href="https://www.openalgebra.com/2012/11/multiplying-and-dividing-radical.html">Multiplying and Dividing Radical Expressions</a><br />
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<div style="text-align: center;">
<a href="https://youtu.be/VbqlsE8iSpc"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjjiBZP8jtTzbRLIZQMHyXVBhts42clsq5ARdf-F93Bn_5YElK8ikb9XKwWasgczg5LQXisUc0iRZSXJKR9dRMtNYSbC9haJVmdtXiYVQYEo_bPU5o9QrPKB1K7gZyGVtH2HbCtNhJDhm8/s1600/vid-0817.png" /></a> <a href="https://youtu.be/IiLYkcUBCi0"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiV8YSfVFgJcKhZiymyFjx6ke5w-qGhOzEnq0yyiBGMxMw7cA9uLhGSu4DMCGH74iipG3eJVHD-3oyaF5sUypZ7LC4HuG3xNxNhKe00h8Ls1vLzAp6lSTkjP5kROACR8-dg-t6FUmp6caQ/s1600/vid-0818.png" /></a> <a href="https://youtu.be/8mXLhc9TqQY"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrbG0rXuJHmHA1lhcyVRBYWlxZm5MmPTQ7BgAaG7EPGypMuzZnMjnhI1xwriKtYxHuCl_lSfdc5NC5hRwKLiKbHN5mqAbf7SYTBRb1Ml-nxGbAYis_eEhmBBHi3heXywdmjsRZiJm8Alo/s1600/vid-0819.png" /></a></div>
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<div style="text-align: center;">
<a href="https://youtu.be/qht8nT6g6XM"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZ6JS4Eb-WgHjtF3w0XYz3tjn_RAQOWEDZHkbK1D6cetQsnLzmCRbVIqYDXwDd8ddLiwvREBWfoiX4LpCFKYNeNj94LDnNDAv18wK09o9hpuWMysdpu8Zc2J4z0oBftdgn5aMc9hCXcmg/s1600/vid-0820.png" /></a> <a href="https://youtu.be/OiS1J2fR9Xc"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh2Vigrs2TWfsSCn7qoJdI6-E0-Jl9hJhBaGBGKGVHJS-p702tTB9KwIiXDjiw_b58GOfG1z9csWq4hNcKeqFtsQOK-tjJMTGq4Aeaa35UzofvWvp-tPrcLq71ZSqL4eOIDKS0sg8kxs5E/s1600/vid-0821.png" /></a> <a href="https://youtu.be/mS3pDb_JerE"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEik6kxNLjoLPSR0sJkw5uYeQz2JfUG7PRzSYByP6wQemPFh3LWs8VXpmFBRKNHZO_xeJcfPcHmV-m_IiOjoD-vFysdY5F95eKGZ297zbuLcTnll6H_J_eLbu4ltTLVf9QJZ5SxEBDl6ijc/s1600/vid-0822.png" /></a></div>
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<br />
<div style="text-align: center;">
<a href="https://youtu.be/B8i-9rZ4l0Q"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_1DeNoCngYK-StPtSTvdNh78mRE0nl7-p4HBg4dP-UHadbS28J2-wRI_YseO4z4wD4CsbyLxuDWswxsbvrUprPov4q3V-fPgR_j7dc3gm80dJhD2pDVveqpjNvCnVuQu5Zoj3WZ3q_0M/s1600/vid-0823.png" /></a> <a href="https://youtu.be/nmKmWbXPwjo"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2NeZTxvoiixGC8eWndOgHlXBLm32aAn8nTHOKOL0wTx5FLUO8foD2S5VAPtIC3vC2-923LiZr5VHcqVfMN-Xo9Ysikyj6821HOhto_heQswOTo9QlNozKoB0nSK_jMHEZmqf2ez1GLAg/s1600/vid-0824.png" /></a></div>
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<div style="text-align: center;">
<a href="https://youtu.be/TPUd_gaJDNU"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhLoNs4GIQdccocex-xVaqD6iDqAuF33ez0oQWj86pa-i6120qfQnYzcOV653dvdhX9iJ9wCrv-jWK5M_hyESK7XxjbuOiBRCWOek8HPaIHb3ohytNMXzO9UPManu9EPCLXNizctrHZHXY/s1600/vid-0825.png" /></a><a href="https://youtu.be/scdd3Xfdpeg"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgHKcU3mNDUUxwGc801JjHpVnTlJAzBHkjhyphenhyphen3UjT051udyZ5Ohjyhdtow0dmsp_3fhIQITV2beOZ7b7vIVc-ZXYiQv4F_TKG-e7zHF8WFtDNkSw_KTNzj87W9V-y_4Y8wBdNZIMR0zMuiU/s1600/vid-0826.png" /></a></div>
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<br />
5 <a href="https://www.openalgebra.com/2012/11/rational-exponents.html">Rational Exponents</a><br />
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<div style="text-align: center;">
<a href="https://youtu.be/uPMspltvkyw"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqyce9j7I5LEq69QZxQ7vUytHb4KRBeAEjNhvho6uHTuggycV6yIL_SLaiD80IhcNr1_tuphO4IJhGCWHVgdXoJ5IYMiCXUOHoaXreOClXBu7mH2Y0JClVlw8sNYVWsKn1N-zumVW-HGY/s1600/vid-0827.png" /></a> <a href="https://youtu.be/WjnbUSMu8nU"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjdavRMSk86QDuFV00Vvt-uV2g03q1I5HRgKvsYUW7uEc1xJnv3eH2DJa3JgnMBTR5MMDs_OaZpuVpgacFTYAoGUHCGntI8f1C3qpUkf3QLXClkFypuU3z43Qypheehn3GxoilhZ7YHw2w/s1600/vid-0828.png" /></a> <a href="https://youtu.be/7iaDthlqI78"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiSJNF8IaheszwiOcJHeLxUjtFpj2su0me_8xA29n0QmpApK8KTBkeV42JlmOz-nkjVsnpED3Jq43y7SvOX48Bxk_A7O5NTogEqbi2ralz36eaOAB2rIbvfE29t0wh3n16nM8InPFTfLB8/s1600/vid-0829.png" /></a></div>
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<div style="text-align: center;">
<a href="https://youtu.be/j4yU4jc4s_A"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiScEQz1dqy7_r-s6My_K7w06b93zDyXckV6iata6tUX2LDYzPLG2BwT9L6TylEa6WhhW05iMMDSblJhz5c8ax78CbeIdpkUGCu4mNr8PkN2MPPidNSUeQgVxiGlnP0D4uTRO2taCR7bz4/s1600/vid-0830.png" /></a> <a href="https://youtu.be/_maq45p4NWE"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhYnWyYGZTzv5kNavaYT4t1JnL6PCZnz63Kjo9zwDz7z0jww4aF3qE7sZj0ko8Zd_IWBqNs4vN8yiHKAO-97GQZ3D3G_uw169meB1EcSQ5KglZfqO5u9AYGPtEmPuRD3d7kPYiw0Pl_k1U/s1600/vid-0831.png" /></a> <a href="https://youtu.be/84ebXSjpcvo"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiFapMw7KhsFyKWASp9pvtbFV_CcfAyv8R0K8o6nAEbAV-ITpANeMIJrBWpPL_mYrZhyfnSce33LrIcPWtqUQBOXkL1l2epw5HbBXTNACnUxylLrK_IFuB7NIVPbsolIzl8WO5z7g06CJo/s1600/vid-0832.png" /></a></div>
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<br />
<div style="text-align: center;">
<a href="https://youtu.be/K4aUwI4JMmo"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjfSXXP1QKFYa6PeLf638m7abjsWVetSmTMPhmWsYnsXee9n1hZBO3ZnXRyXal1BOS5Bq53MoTs4q9rUcGjwZSiRJp6vSBG0VoWL6nK2Gvrkw63O06lwSyKLWx-mIR8912VFYBHlZQPI3A/s1600/vid-0833.png" /></a> <a href="https://youtu.be/kPAgqFrByFA"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUxo9b-_bhSWmWGxfoBFizPh_96pWhwP2OzGyNPwoakdd20bTX9_zUa1qsOVSjWeOhNwZjZyH2vhFQYjLKzAO2hL-pnNFgX_nXuYCOSLbgqTot8vGXRCIzdGdcjXE-1JDEukE9XJj-GpQ/s1600/vid-0834.png" /></a></div>
6 <a href="https://www.openalgebra.com/2012/11/solving-radical-equations.html">Solving Radical Equations</a><br />
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<div style="text-align: center;">
<a href="https://youtu.be/zlMHq08Adgk"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgjQbVKfTFqhXDH78TcCCMVfTPIK7qk6e4FD99H_zv9LTqxdsbcADZ8EojeIhw5zyIYIeN4jBVpNqs9MsCvWAgMaW81_KuYuENXppBvhA4LmiyVZROPS_vlPH6bZ6RO7vrvCw0Y1J_5jeA/s1600/vid-0835.png" /></a> <a href="https://youtu.be/bY6DT-X9hTg"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhPjh2orVx9I1CnJYS45tIKnJ5b7ANT-z_UeVAeyP9nQnz_9b3qJf2YizEQOwn8UANHwGtnWZyfvq53O4_mZcaZtMTkKf0bWQsM1UovvQd05P35WoRAdkOSrPxJUm7yHaweZNzqWwwF1Us/s1600/vid-0836.png" /></a> <a href="https://youtu.be/cneOXvx-n4I"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgQdWrM9LhX-FOxvGBcPSs1dFOSRPPFPSZCUVDDME0uqF8Rzdemaeci0tyBty5fakvSH6FZl1M3QHi9pZwwLvpbKIbyvqCWGLPhnjwBD9MICNHnuaU3ZK8hiNCV3cCtZTILQw961iIl5ak/s1600/vid-0837.png" /></a></div>
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<a href="https://youtu.be/9icCm1p2clw"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj7L1gWbetk7lcvbpJmNJr6PUbHJ2jUgluszajtyjBcBycdrrmwE38gk3lvqCAH6qU2lfEc8mpaSNxbk0oGKco2vtOWEvjriTVCa6dR6hZFLxDF3rRpH2xQmhyLXVy0ODNk4-eaMAiDxdA/s1600/vid-0838.png" /></a> <a href="https://youtu.be/O87kFST_KtY"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEizqpJTeIIs3qfYfIwdXzYVi2W03GkD-AjNj_3xbfm0h9RA-87xmBYMA6To_4xTWFpANu0g3qRPHdK4Ct-hHDWWENROtiuKruhhFIxWIm6MRyumniVN1aZp2oJGn06dM-NYqo6NGtvdMlc/s1600/vid-0839.png" /></a> <a href="https://youtu.be/0tYFLH2cziY"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgCxajmMI70E-L83teoFf7EVwj287Vh1R4spWngzeiVC78GyNWNXx5ZMTRdA9ZgzW2CgP3zwh6ylq2eUGU0w7nY6A8_6vsoUT8A6RdqkV8J_WXKf-VCAFKSEWbe9ChKtXmkstmKoG33zBg/s1600/vid-0840.png" /></a></div>
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<a href="https://youtu.be/kJMALAnb_I8"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgo3nNZ54kFst0cYj0HfxAGlY8mR9VK8voQWwE6iW0QW4sl-Ql6Fex6E41LI253ObOAM4ShcRr2luiTHNcRxs4lhaIVg8gY5miByDsf4ezAGemIEylsLdyLeSBuR7VvbSYeoWviyOagVdU/s1600/vid-0841.png" /></a> <a href="https://youtu.be/4AmAZNCrEn8"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEheFJB5DZx2iCutjtvkri-_RJhdy2xWwnQ28ZROPB3syfkefEDVJv0G59fLnPq8-31n2KoYdUWXNIqi2ZB1qZrjC3KBDVLxyf9chyphenhyphen7VFcX5oI6p1iANzGOt5aGHhW5IDRkVxTkN5LnbQgw/s1600/vid-0842.png" /></a></div>
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<a href="https://youtu.be/3nuwmnNaHdk"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhqEaaj9LC4AXnZxyqtwLokeMzRQcCTc0JzypBPrmFszjpsEkIQdSzEeF2mS22Xjlq3Q0TDcRhuLJ07LQ-WVeZFq3JgYSesHG_bMxPkMJ1ERkR0z4SyAmHD7gCKKhRgbh2HdUO1oO0nI9I/s1600/vid-0843.png" /></a> <a href="https://youtu.be/p5OAU6EscxY"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj8qe_mG0dJiA35RJcVkYLca866fS54xarOHoqpWVXKxrKFLhsxInpfY1iwM2VGzewwid9rdH0GP3xit8pO8_sVtQQFuGpLBu3IkG2Nfv5HRR38gLU_P7fYPMmJLuNkyu7lePu09wJstTg/s1600/vid-0844.png" /></a></div>
8 <a href="https://www.openalgebra.com/2013/03/complex-numbers.html">Complex Numbers</a><br />
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<div style="text-align: center;">
<a href="https://youtu.be/hUP3xpXU-yo"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEie6e9NPSzn518973xmcMFWJT4zLgV7wTLUC4DVcvpbtgUGOdi9Yd1q2Ssr09xwNtKwvdKgeXwEOeovhEzNH3QUE5oLb_LCVL3ym4R5W1JDrBTpThirgTvZuOUwJeoeCI6US1XhMb3Y6fM/s1600/vid-0930.png" /></a> <a href="https://youtu.be/kGcCLQ-qQ7E"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhHMkB0Krsg3AHdFoQwlqNMOr2QlNFOdE2moeCo39VvWo2LKcZpyZIDCE0pQcaM4WfNEdJyEQ3m5nZi4EkHgqOgG59PhaNaVXJYkmfmjjB76CikPHokd8dWqc9ccAYNWcRkTMDNsZYYnf8/s1600/vid-0931.png" /></a></div>
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<div style="text-align: center;">
<a href="https://youtu.be/vKLrDLDaBEk"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2TCOjwm_Deu-X1xFnMY4EFOQ0PV1vmlXHYjf8JcJVw5k3_AIJm3vWQcW-JWRdKaO7ESZ4nhneUraT1hRNVJIvGvfhSYbG4Q-vAQSX7tN4FudKPE48ln9qfGX2UvpFOpXZwEXoK9b7hII/s1600/vid-0932.png" /></a> <a href="https://youtu.be/UKJ6p2jgT-Q"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEim5SrvmlSeBY3G2TwXvgOO_npEDsjqTshMxv5gLE5gKq-1jFlwCs4LqhZ_LqrP5hBrc1ibyco-BwyCJbwiYOttFajfXvzzJOUBxRdU3Vb2y7_MON8LBkp_UpxU_3vJpxFQ24xvbgPTy58/s1600/vid-0933.png" /></a> </div>
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<div style="text-align: center;">
<a href="https://youtu.be/ruQnpTadueI"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvDYm5fthnhFbkMXtrkTPTokoyoQKYBAi1SMkvN22JIY6h1KuOaYhZvbGinoVU2oanCic72LQwh5m3A1O3iNGXaScHAQGUPti8jH3A1DeRcXo6R0tlOfEb1BvbwoiX07UPNsfwUokjj9k/s1600/vid-0934.png" /></a> <a href="https://youtu.be/clXnT-ogmK0"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEihk4050XJQZ83TZLxHLtzzou5S6N41cCJMWY9NbN03yQjhmVCmrAyIBvhUAd4SVbD8QT14Mz64FO1KShq9fCY25_WAMztJ70TSDI35q7LBZ8XGi-QlK34_AXTvyz1GuJ8WqP-v1jgFlkg/s1600/vid-0935.png" /></a> </div>
<a href="https://youtu.be/ruQnpTadueI"></a><br />
<div style="text-align: center;">
<a href="https://youtu.be/zyLsmo223SE"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg35mEoxwEGishMLlJppY7ylpwb__iVDrniHm9Qw__MX3MGpvTu_zS0rmw24x_WCcY_HabfRq9MirD2Da-o2pm8MoNtFGtnoYJXoRrLLyN27JOoi-Qrx-9t7FEXPgQhsodHcJuZoh62OX4/s1600/vid-0936.png" /></a> <a href="https://youtu.be/k3S4_m5eb7U"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhSAPPtNppeJrYnhCyXHVTi8WV641neAgwYMhKnVzEgysUa7CoE5I44t-Tr6Ws2VsJgpCJh0IX5K9GXA1j3yo9cobOwPdhgETKGzN107JnyhKHqFvsEBqKF6PJONDAmElm3PqW02huVgkk/s1600/vid-0937.png" /></a> <a href="https://youtu.be/_kkdM6z4320"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhqkSAwGqqqdTIWmiuc74zEVpbpTLPR1-3gM2uCJNgIjtKA_7qJiaQbcXzi1Noinxpb16ThRJc937i7BHHqXU17gqJpa3JKWGE948OQ0OkT3MCbDBex-EU6nm5KUneUkcp_LpND3wfXzJc/s1600/vid-0938.png" /></a> <a href="https://youtu.be/ck7adtbc55o"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh2lgHN5IHlyVdgO0SQrO6UxVkoJht22_1h39EYTl0nbzs0zff_6DCgbQOExtVeXsK0zAjNFa92AtbKnD4djDm2r-bfe40QdDIexvoFI3eg9ixQI-ab8bChlrLgRcgO4F4lggW5sBTkg9M/s1600/vid-0939.png" /></a></div>
<a href="https://youtu.be/zyLsmo223SE"></a><br />
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medium;">Tracy Redden's support course.</span><br />
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1. A review of graphing lines, finding slope, and finding equations of lines from 2 points and a perpendicular line.</div>
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2. Review how to multiply and Factor all types of Polynomials. I cover Common Factoring, Factor by grouping, Trinomial factoring, and difference and sum of cubes!<br />
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3. I show you how to solve all types of linear equations from basic linear equations to more complicated ones with fractions and variables on both sides. There are also the ones that end up with no solutions and all reals as a solution.<br />
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4. The first step into learning how to solve a quadratic is by factoring. Here I show you how and explain why.<br />
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<iframe allowfullscreen="" frameborder="0" height="270" src="https://www.youtube.com/embed/p8epkqdZ2wc" width="480"></iframe><br />
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5. In addition we will look at the domains and restrictions of Rational Expressions.<br />
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<iframe allowfullscreen="" frameborder="0" height="270" src="https://www.youtube.com/embed/zZX-P5BsTvI" width="480"></iframe>
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6. How to Add and Subtract Rational Expressions. I show you how to find common denominators so you can simplify.<br />
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<iframe allowfullscreen="" frameborder="0" height="270" src="https://www.youtube.com/embed/ZVv9IbbWYrE" width="480"></iframe>
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7. How to simplify Complex Fractions. I show you two different methods.<br />
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<iframe allowfullscreen="" frameborder="0" height="270" src="https://www.youtube.com/embed/j-BQXTNCbGw" width="480"></iframe>
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8. Solving quadratic equations.<br /><br />
<iframe allowfullscreen="" frameborder="0" height="270" src="https://www.youtube.com/embed/uvnGNEibiiE" width="480"></iframe>
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... more to come soon.<br />
<br /></div>
John Reddenhttp://www.blogger.com/profile/02231088988057526726noreply@blogger.comtag:blogger.com,1999:blog-221233850938098964.post-1103938196984309452019-10-22T11:08:00.001-07:002020-09-27T18:30:46.977-07:00Synthetic DivisionWhen dividing polynomials of the form <span style="font-family: Courier New, Courier, monospace;"><i>p</i>(<i>x</i>)/(<i>x</i>-<i>a</i>)</span> we can use synthetic division as a shortcut for <a href="http://www.openalgebra.com/2012/11/dividing-polynomials.html" target="_blank">polynomial long division</a>. Below we divide using traditional polynomial long division and synthetic division side by side.<div><br /></div><div><div style="text-align: center;"><iframe allowfullscreen="" frameborder="0" height="270" src="https://www.youtube.com/embed/Enij6-7Ury4" width="480"></iframe></div>
<br />
<b>Divide.</b><br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0JHQAX2k-1bvIjmXiiaQK3LcZNArqDIGHlL-QGxcMpOc57l7M7kKUnirRcMB533Ie-Cr9jV1dwy9o23YXAcWUgBaNI9FOm5CqABASVtr7BGKOhs0T3ze96aTHu_YfRUZDipQnKExuJsY/s1600/example01a.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0JHQAX2k-1bvIjmXiiaQK3LcZNArqDIGHlL-QGxcMpOc57l7M7kKUnirRcMB533Ie-Cr9jV1dwy9o23YXAcWUgBaNI9FOm5CqABASVtr7BGKOhs0T3ze96aTHu_YfRUZDipQnKExuJsY/s1600/example01a.PNG" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgFX043W2w9RDNBCjQ-R6RmVRuwy38oLRVrep531fTI0dEYrpN1i6OtgKoYumds97gyGwwob2HrijQeBHE64zDodeel87rpvUq5TLjDhnfvuTP-ZM5OfInbLllwmDem_xseUsG5VFxjYSM/s1600/example01b.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgFX043W2w9RDNBCjQ-R6RmVRuwy38oLRVrep531fTI0dEYrpN1i6OtgKoYumds97gyGwwob2HrijQeBHE64zDodeel87rpvUq5TLjDhnfvuTP-ZM5OfInbLllwmDem_xseUsG5VFxjYSM/s1600/example01b.PNG" /></a></div>
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Answer: </div>
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Both processes give the same result, <span style="font-family: Courier New, Courier, monospace;"><i>x</i>^2 - 3<i>x</i> - 2</span>. However, synthetic division uses only the coefficients and requires much less writing. To understand synthetic division, we walk you through the process below. Be sure the polynomials are in standard form, that is, each term is arranged in descending order from highest power to lowest.</div>
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<b>Step 1</b>: Write the root <span style="font-family: Courier New, Courier, monospace;">a</span> determined from <span style="font-family: Courier New, Courier, monospace;">(x-a)</span> and the coefficients of the polynomial in the first line.<br />
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<b>Step 2</b>: Bring down the first coefficient and we are ready to begin.<br />
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<b>Step 3</b>: Multiply <span style="font-family: Courier New, Courier, monospace;">a</span> by the first coefficient and write the result under the second coefficient.<br />
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<b>Step 4</b>: Add the second column and write the result below.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh8EJ8uorAKk4xjrMLxtYGx1k0orLmj3TudY3rL6Q0OwKb8zNo8ts329Ce_ErSISc10VuxfEAO-1ihGDEMrTqmHTJlcW6aBvCpAj9ZIl64a0JdpTCjVzuc-8Pn4xGy-KQeYRgY_A7TlmFc/s1600/step4.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="90" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh8EJ8uorAKk4xjrMLxtYGx1k0orLmj3TudY3rL6Q0OwKb8zNo8ts329Ce_ErSISc10VuxfEAO-1ihGDEMrTqmHTJlcW6aBvCpAj9ZIl64a0JdpTCjVzuc-8Pn4xGy-KQeYRgY_A7TlmFc/s1600/step4.png" width="200" /></a></div>
<b>Step 5</b>: Repeat the process for all of the remaining columns.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcfaopCCrGXWYDsL5H-ptH3JF-Qaini6yg71ABq7QbFKgwfnX9G4Ap8NgplHdBdzRlO32E2qyTir7xKeZ5LAVlnaeOv05aQ8H9DGXLL9JIavyAXGXIkN6mrL5uqjH5Xw3ayy27emZtvC4/s1600/step5.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="84" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcfaopCCrGXWYDsL5H-ptH3JF-Qaini6yg71ABq7QbFKgwfnX9G4Ap8NgplHdBdzRlO32E2qyTir7xKeZ5LAVlnaeOv05aQ8H9DGXLL9JIavyAXGXIkN6mrL5uqjH5Xw3ayy27emZtvC4/s1600/step5.png" width="200" /></a></div>
<b>Step 6</b>: The numbers along the bottom are the coefficients of the result in standard form beginning with a term of degree one less than the original polynomial. The last number is the remainder.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_vTii4n2tCa8IBpjL295DzyBjUswcAbwq2zCmfroatGsir_sEKTZbHmZPyXbAO8Dws1N4MqIlomXXEtAq6lbKWyDmUFLkLvHsCclVjuAA3uq4Hzxa0cj4AWPXFkdrHXt0QWYcN06WEv8/s1600/step6.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_vTii4n2tCa8IBpjL295DzyBjUswcAbwq2zCmfroatGsir_sEKTZbHmZPyXbAO8Dws1N4MqIlomXXEtAq6lbKWyDmUFLkLvHsCclVjuAA3uq4Hzxa0cj4AWPXFkdrHXt0QWYcN06WEv8/s1600/step6.png" /></a></div>
To finish, clearly present the answer to your reader. Next we do an example with a remainder. Just as we do in polynomial long division, we add a term that is the remainder over the divisor.<br />
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<b>Divide</b>.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjonCfj96cquUY0kwqJTn2le7AJoEu04DB05uarxfZY9UNX3crF6KyEY3LG9cdV-jAKpsdYX6Mzi_P6P_KxJVFn1a6Q0jhLcec6Oe-TrO1_gkQPDcCnk-X4TCXkbgo28jU7XDfI9rUNWWQ/s1600/example02a.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjonCfj96cquUY0kwqJTn2le7AJoEu04DB05uarxfZY9UNX3crF6KyEY3LG9cdV-jAKpsdYX6Mzi_P6P_KxJVFn1a6Q0jhLcec6Oe-TrO1_gkQPDcCnk-X4TCXkbgo28jU7XDfI9rUNWWQ/s1600/example02a.PNG" /></a></div>
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Here the root (or zero) of <span style="font-family: Courier New, Courier, monospace;">(x+5)</span> is <span style="font-family: Courier New, Courier, monospace;">-5</span>.</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiiJ70DJyEYsYsl7w6lTCyYxxQjEU7G75gvEs3aqsb5J9S7-3hi6JZb2Gg1ruozbu-XxEWoZpUA8j8c78o8IY2tWT8rnq6eBglmeT1nxQEeY9Swv12zgITwnQko7Mr6tHgz38F3XD-GblU/s1600/example02b.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiiJ70DJyEYsYsl7w6lTCyYxxQjEU7G75gvEs3aqsb5J9S7-3hi6JZb2Gg1ruozbu-XxEWoZpUA8j8c78o8IY2tWT8rnq6eBglmeT1nxQEeY9Swv12zgITwnQko7Mr6tHgz38F3XD-GblU/s1600/example02b.png" /></a></div>
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</div>
Multiplying both sides by the divisor <span style="font-family: Courier New, Courier, monospace;">(x+5)</span> we have the following.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiXMpYrusf82dG2mvFEI4NfMCAj56LITe6VKz8RK12XHcfrkEasM4apmWVVMn2KpnWDV3LPFnxYP69Ubd-vSV5hW4eM7Qjox_P04yXQZwmxJ0-i9t_O9FsJglVJMBDN91HxkCDvyJgIrJs/s1600/example02c.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiXMpYrusf82dG2mvFEI4NfMCAj56LITe6VKz8RK12XHcfrkEasM4apmWVVMn2KpnWDV3LPFnxYP69Ubd-vSV5hW4eM7Qjox_P04yXQZwmxJ0-i9t_O9FsJglVJMBDN91HxkCDvyJgIrJs/s1600/example02c.PNG" /></a></div>
We mentioned that the polynomials are required to be in standard form. Sometimes there will be "missing terms." That is, not all powers will have nonzero coefficients. In this case, we use 0 as placeholders when performing synthetic division.<br />
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<b>Divide</b>.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhw56EkqA6CDWYfOIMKlKF4Z_OCw8fy8vPOtpyTCu75OIt9faaArRarq8ikeJJs_dGdc8H3Ryh6hGpyBJtg4MARsGN1jm9EhY9GZRklACIl5BCI3aOqazr3XPJgbjrG5XiZakr2Nptvhmg/s1600/example03a.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhw56EkqA6CDWYfOIMKlKF4Z_OCw8fy8vPOtpyTCu75OIt9faaArRarq8ikeJJs_dGdc8H3Ryh6hGpyBJtg4MARsGN1jm9EhY9GZRklACIl5BCI3aOqazr3XPJgbjrG5XiZakr2Nptvhmg/s1600/example03a.PNG" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiyHE4-aauOswfbxXhc2_QxuCPXMwmu5zJaAkrtmuPurod-uKPj5cPtAEv3MErFdcEd1D-JzEuovVkD-I4acWrcysMno3B50_WeCtf09Rou9nob8XeT5stmS-98pL-vFoTqgIYg4_svq-8/s1600/example03b.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiyHE4-aauOswfbxXhc2_QxuCPXMwmu5zJaAkrtmuPurod-uKPj5cPtAEv3MErFdcEd1D-JzEuovVkD-I4acWrcysMno3B50_WeCtf09Rou9nob8XeT5stmS-98pL-vFoTqgIYg4_svq-8/s1600/example03b.png" /></a></div>
<br />
<b>Answer</b>:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhobc-tVA-l193zHjqbB-zigzYQFn2uGJjvHE9AliOf-yy5942ICTMrLXjYu3BqEveZwBhFdNmuGYI7OWBdeTevBjm0SHaqSr2il0ioRnHNgFSC66Mo2OjF3cug6j-sxaG87A_pfGd_40A/s1600/example03c.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhobc-tVA-l193zHjqbB-zigzYQFn2uGJjvHE9AliOf-yy5942ICTMrLXjYu3BqEveZwBhFdNmuGYI7OWBdeTevBjm0SHaqSr2il0ioRnHNgFSC66Mo2OjF3cug6j-sxaG87A_pfGd_40A/s1600/example03c.PNG" /></a></div>
Sometimes the root will be a fraction.<br />
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<b>Divide</b>.<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh2BN4NdEcBgddikqSn-8fqIYzCrtilloxiXOQZKWj4e6mJJTTAqw5gIoqSpWLeR1bwV8WF1SvuV0lhSUoQESrh9TsD40eiG1I0Rf_DXpV-5wPBBAmGki_IPEx9x00MkT7Xc-cK1c4ZrQQ/s1600/example04a.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh2BN4NdEcBgddikqSn-8fqIYzCrtilloxiXOQZKWj4e6mJJTTAqw5gIoqSpWLeR1bwV8WF1SvuV0lhSUoQESrh9TsD40eiG1I0Rf_DXpV-5wPBBAmGki_IPEx9x00MkT7Xc-cK1c4ZrQQ/s1600/example04a.PNG" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_fB-IhfCKgcPEKxJ3h6N1lITVECVO8tcGov7wsYmvl3Y8cILjfkzP7Gm3yCejJD_NiwqrxObsjBK-i2rzkUfR3p14o6V-wBdXm927SN8b3gOsVYUoa5YgDPramSx6TuH8ltlIfV2q5xE/s1600/example04b.png" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj_fB-IhfCKgcPEKxJ3h6N1lITVECVO8tcGov7wsYmvl3Y8cILjfkzP7Gm3yCejJD_NiwqrxObsjBK-i2rzkUfR3p14o6V-wBdXm927SN8b3gOsVYUoa5YgDPramSx6TuH8ltlIfV2q5xE/s1600/example04b.png" /></a></div>
Answer:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgu9bok8ewnEnbjSD0XBRSQ1HWhDGwMlMJtrbo2NTh4fSCnNRH-gkIvM4MjJZETjgNl_RJPxyGOtnvQ9QVCQPf9mc7tzVW8odDaFFQRm_rRz0FCBJ6v0wu8KVXhrIdGTxysr1vV7F-N2IM/s1600/example04c.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgu9bok8ewnEnbjSD0XBRSQ1HWhDGwMlMJtrbo2NTh4fSCnNRH-gkIvM4MjJZETjgNl_RJPxyGOtnvQ9QVCQPf9mc7tzVW8odDaFFQRm_rRz0FCBJ6v0wu8KVXhrIdGTxysr1vV7F-N2IM/s1600/example04c.PNG" /></a></div>
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It is interesting to note that the result has a GCF of 2 and we can do the following algebraic manipulations:<br />
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhIlLlAcjY7qY8Nn91xNOLE4iC5LJuaZZnjLESEoUrhpqsxWy5lc2yEX8Zqp_BYxN1swUYCDJFfalbdwaFWmP4G6BP2dwm2mSuR3g1Idl7EUAW72m0StAQXFe8QepSb4msdV1YETQVN7gk/s1600/example04d.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhIlLlAcjY7qY8Nn91xNOLE4iC5LJuaZZnjLESEoUrhpqsxWy5lc2yEX8Zqp_BYxN1swUYCDJFfalbdwaFWmP4G6BP2dwm2mSuR3g1Idl7EUAW72m0StAQXFe8QepSb4msdV1YETQVN7gk/s1600/example04d.PNG" /></a></div>
In short, this gives us a method of factoring a more complicated polynomial.<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEghtMVLlj8J_sxHB_WDvwihttwI2UWz3wq1BwtG3eJhLG1bmWQGKG992OlShZ2r5OYkWrh6Dvt2r10CQezQYzVTgUyPbBdU34MBIhmQ5lq8HEhbtcj8ogms8CYDiwdm3l65uV3SbL1VIAc/s1600/example05.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEghtMVLlj8J_sxHB_WDvwihttwI2UWz3wq1BwtG3eJhLG1bmWQGKG992OlShZ2r5OYkWrh6Dvt2r10CQezQYzVTgUyPbBdU34MBIhmQ5lq8HEhbtcj8ogms8CYDiwdm3l65uV3SbL1VIAc/s1600/example05.PNG" /></a> <a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQOohfPUpPqqQv_IejemQkHMyYwGnqyOHzCqxGTwWqoR069I76mHu39KM2WU19SEAZ7DTgMResMkfRAeccVwuJiE-o7TLIHPMbyN_ruK0T_s4IpR2bogJupyGPddTbRKESR_iDxZcTImQ/s1600/example06.PNG" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQOohfPUpPqqQv_IejemQkHMyYwGnqyOHzCqxGTwWqoR069I76mHu39KM2WU19SEAZ7DTgMResMkfRAeccVwuJiE-o7TLIHPMbyN_ruK0T_s4IpR2bogJupyGPddTbRKESR_iDxZcTImQ/s1600/example06.PNG" /></a></div>
<br /><br />
---</div>John Reddenhttp://www.blogger.com/profile/02231088988057526726noreply@blogger.comtag:blogger.com,1999:blog-221233850938098964.post-58245186805383475572019-08-01T19:14:00.000-07:002019-08-01T19:14:00.910-07:00Graphing Basic Functions with Translations and RefectionsNotes on graphing functions using transformations. We cover rigid translations and reflections as well as domain and range.<br />
<div class="separator" style="clear: both; text-align: center;">
<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh0WRg0s_3x9GpHuKkrLAbA-8SKkUcPhkvhiG8vSkccw1EIkvnUByUQTdRAI0WGYN8ENIxAK6EUE3GEC287TE56cgmTLWD5MCfc7htCzzrZr8GqkxeyVilHG4AW87hG7wxN08UNkniFqvo/s1600/VLTanimation.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh0WRg0s_3x9GpHuKkrLAbA-8SKkUcPhkvhiG8vSkccw1EIkvnUByUQTdRAI0WGYN8ENIxAK6EUE3GEC287TE56cgmTLWD5MCfc7htCzzrZr8GqkxeyVilHG4AW87hG7wxN08UNkniFqvo/s1600/VLTanimation.gif" style="background: rgba(0, 0, 0, 0); border-radius: 0px; border: 1px solid rgba(0, 0, 0, 0); box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a></div>
<br />
<b><span style="color: blue;">Notes:</span></b><br />
<span style="background-color: white; color: #444444; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px;"> 1. </span><a href="https://www.openalgebra.com/2013/07/relations-graphs-and-functions.html" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; text-decoration-line: none;">Relations, Graphs, and Functions</a><br style="background-color: white; color: #444444; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px;" /><span style="background-color: white; color: #444444; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px;"> 2. </span><a href="https://www.openalgebra.com/2013/02/graphing-basic-functions.html" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; text-decoration-line: none;">Graphing the Basic Functions</a><br style="background-color: white; color: #444444; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px;" /><span style="background-color: white; color: #444444; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px;"> 3. </span><a href="https://www.openalgebra.com/2013/02/graphing-functions-using-transformations.html" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; text-decoration-line: none;">Using Transformations to Graph Functions</a><br />
<br />
<span style="color: red;"><b>YouTube Video Tutorials:</b></span><br />
<div style="background-color: white; color: #444444; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; text-align: center;">
<a href="https://youtu.be/uEtVDkjVSX8" style="color: #4d469c; text-decoration-line: none;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh51B2qhybqz8PQoErC-CkezeWConwWb0z7Y2tGOi-iPHVlUt9285FARaQj-UvyRSWuf9Xm3DMVjbzDIluguGDNqLIs1GoJjorrHfs99fpz6TRRLAuuIiVrSnmrp-G6y7YY-LXf4kBbGj4/s1600/02-0101.PNG" style="background: rgba(0, 0, 0, 0); border-radius: 0px; border: 1px solid rgba(0, 0, 0, 0); box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a></div>
<br />
<div style="background-color: white; color: #444444; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px;">
<a href="https://youtu.be/Tum4acbD34U" style="color: #4d469c; text-decoration-line: none;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgfc8IL5Aeg6pTGecoYuWDB1BnJ8A4yG6b3hnSiJU71B-ffAT5gWvNwLUzPKFkEu1oCerQ1loZ9qbQwMBQyO2JnJ3rq2W_InUMHu5GMI1izyL9kmrbIoKHwXDRXg614nbTyraejMUByOzE/s1600/vid11-4-3.png" style="background: rgba(0, 0, 0, 0); border-radius: 0px; border: 1px solid rgba(0, 0, 0, 0); box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><br /><a href="https://youtu.be/k0EzHLP26bg" style="color: #4d469c; text-decoration-line: none;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjo-p11_JZ0IiNplyZZ1QKUhtgUsLG9hI47xipjo6R9_CFPGxKWpnIAV3um2tnnyIMlWG3zrOafUmBwS2bOFsss-F0bUvtL4dbqt3W0_bQ7TdYz5i8hZt-L0CqkT50vytz3AXPyXvotl4I/s1600/vid11-4-4.png" style="background: rgba(0, 0, 0, 0); border-radius: 0px; border: 1px solid rgba(0, 0, 0, 0); box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><br /><a href="https://youtu.be/daAbQSam_MI" style="color: #4d469c; text-decoration-line: none;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUKRnPt9mZBJdwc9qciBQ_v80SNqyRqEbxKEBFCIDlolKd_P6Fh8dx7rqAYt0P1RaP-i8FZGzFUWoFuJreLLvSw2cOoxVomsePCD5ZgbHGKRSHidCBpA41zsGOOmu-52_n-2pOiwxIjLE/s1600/vid11-4-5.png" style="background: rgba(0, 0, 0, 0); border-radius: 0px; border: 1px solid rgba(0, 0, 0, 0); box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><br /><a href="https://youtu.be/kA1doxxQTHs" style="color: #4d469c; text-decoration-line: none;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_h9Ee6DOzuM2bu6ICtGfaQheCKs1IFXdAYO04rmC03cbdsDEyL9O9RJD6Ks2cegoUVThIXfab1SlXPQUA7fnTDoiojW0AFfnJmmF_p0zFrms1inpP4fbSONLw11zFC1zT4iyYlVFfg-Q/s1600/vid11-4-6.png" style="background: rgba(0, 0, 0, 0); border-radius: 0px; border: 1px solid rgba(0, 0, 0, 0); box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><br /><a href="https://youtu.be/JVAb5Bt198A" style="color: #4d469c; text-decoration-line: none;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigwGA2VaJrsVGMVq8ZbAmw7zc2iML6WkxHxjqqzlzSGliD7XkJb-CT5lchY2FjDDcNEe77MrDi66pg2q-x7KTFTYSJYAFp5g1Nnf9YOpq1hlUaCir8xld2eup22MB2oaYfOChYNHPUmeY/s1600/vid11-4-7.png" style="background: rgba(0, 0, 0, 0); border-radius: 0px; border: 1px solid rgba(0, 0, 0, 0); box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a></div>
<a href="https://youtu.be/l69kqACYufQ" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; text-decoration-line: none;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_CTuyhMs8hJ9UsQ3sVwG9U_uCZYdI42fQgMPRXzAD19xTlVvTCFwFxLXSuCJLNe7AX-8xphCUcfYevxM324Ic5gL8Jr6Qkz3G_quSIpSjcOaq4XmBFb19M8mXsB2Pp5_G1KKT1abmfBo/s1600/vid11-4-8.png" style="background: rgba(0, 0, 0, 0); border-radius: 0px; border: 1px solid rgba(0, 0, 0, 0); box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><br />
<div style="background-color: white; color: #444444; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px;">
<a href="https://youtu.be/X9aa4qawHt8" style="color: #4d469c; text-decoration-line: none;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUhFNibYKgaJoGHMpMvhtATcUq24z4EvoR4y7nGXwjN4rMys3GWbtRfuiEZXzoqZeu1td-wm1qKJXurnWPN9SA6Viz_jTvCTGuKittowdCU9TNe2dWPI7pm53lRTeiFn61xQqqAf98vw8/s1600/vid11-4-9.png" style="background: rgba(0, 0, 0, 0); border-radius: 0px; border: 1px solid rgba(0, 0, 0, 0); box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><br /><a href="https://youtu.be/7osHH-Fkces" style="color: #4d469c; text-decoration-line: none;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj5MBFXs1EouDiCbVdIiJOOvqh3fCtKd-cBPJ3M1dRRmh7noRfSEFI2E7pj3ETxCn37hbAXNGg3tzJ6tWSKwRbbxMaHHbCZcWxXYMdBcWf4BOsktv-Zq5LcmNXuP7gU0_0CqVZpXCTctAE/s1600/vid11-4-10.png" style="background: rgba(0, 0, 0, 0); border-radius: 0px; border: 1px solid rgba(0, 0, 0, 0); box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><br /></div>
John Reddenhttp://www.blogger.com/profile/02231088988057526726noreply@blogger.comtag:blogger.com,1999:blog-221233850938098964.post-75070879554816596712019-07-31T16:43:00.000-07:002019-07-31T16:43:07.263-07:00Solving Absolute Value Equations, Quadratic (Polynomial), and Rational Inequalities using Sign ChartsFree math notes on solving absolute value equations and polynomial (including quadratic) and rational inequalities using a sign chart.<br />
<div style="text-align: center;">
<img height="114" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEghjSS7nV7ybNy-r8kCjMw3qxw7ftRDccOawfSniZUzVkqLbs0onrlTxLAkm3ZsLW_D7GQkfrFmIgDkUyMRiJbYyxJ_KjFE-PulOxp3Twc-TXGKNDguXRFBJ9HtoJIeI_YrnVDNzzHGWP08/s200/example8c.PNG" width="200" /></div>
<b><span style="color: blue;">Notes:</span></b><br />
<span style="background-color: white; color: #444444; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13px;"> 1. </span><a href="https://www.openalgebra.com/2012/11/absolute-value-equations.html" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; text-decoration-line: none;">Absolute Value Equations</a><br />
<span style="background-color: white; color: #444444; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13px;"> 2. </span><a href="https://www.openalgebra.com/2012/11/absolute-value-inequalities.html" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; text-decoration-line: none;">Absolute Value Inequalities</a><br />
<span style="background-color: white; color: #444444; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13px;"> 3. </span><a href="https://www.openalgebra.com/2012/11/quadratic-inequalities.html" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; text-decoration-line: none;">Quadratic Inequalities</a><br />
<span style="background-color: white; color: #444444; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13px;"> 4. </span><a href="https://www.openalgebra.com/2012/11/polynomial-and-rational-inequalities.html" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; text-decoration-line: none;">Polynomial and Rational Inequalities</a><br />
<br />
<span style="color: red; font-weight: bold;">YouTube Videos: </span>Click on a problem to see it worked out in YouTube.<br />
<br />
Absolute Value Equations:<br />
<a href="https://youtu.be/OBmDWJ7Bt1s" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; text-decoration-line: none;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj0hNzNi_Ls1CfsJEtYanqK0Gakpay8f2v08zKohpFCCrOWSPt1WPxqQ1PEZLFKWGUHrmJgsXwkJiJ0CZYVI0HbrjrknszUw9baPCbBzPuhmO-dNsgsPKTK9NOwqYz67_TH6tw6WA1GIXbN/s1600/vid01.png" style="background: rgba(0, 0, 0, 0); border-radius: 0px; border: 1px solid rgba(0, 0, 0, 0); box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><br />
<br style="background-color: white; color: #444444; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px;" />
<a href="https://youtu.be/NxuzN9lAkhY" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; text-decoration-line: none;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjqPp-DvFxYKJFRkbbLy8H00czHTN2-zc5Ky-dBwrOMGBmHB7eWb0imLVpTToPLFQrKUpfsZsq0MsHB6EHQbeudwKWAzwkXKVdiVgGQiOq-x10rIENB4VsSt2ETdpyPWb5pLi3g-m0WM9iW/s1600/vid02.png" style="background: rgba(0, 0, 0, 0); border-radius: 0px; border: 1px solid rgba(0, 0, 0, 0); box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><br />
<br style="background-color: white; color: #444444; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px;" />
<a href="https://youtu.be/n45i2_cmAS4" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; text-decoration-line: none;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEha4ER2sdfoz8FsGOjp3e10ZIffd-Yd7I33CDqNApKi19PX25aQYlZd1ROMg3WoJkOm47PxeojrW2q-w_TGiW-Otx2qtH4WFC1I_RIZUt4kJW61Ern0p9cuzLDVhQ6kUTjrVdrX2WjSBQES/s1600/vid03.png" style="background: rgba(0, 0, 0, 0); border-radius: 0px; border: 1px solid rgba(0, 0, 0, 0); box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><br />
<br style="background-color: white; color: #444444; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px;" />
<a href="https://youtu.be/KmNGiafNmsU" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; text-decoration-line: none;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgBPIJwqx3n2XdTsDi0Fj_u5S05MLakhoKA1hLDMSw27ZJIunj6NUReN8RwTgnqFi4TK1w1LYhXXMt7yNVbRJ0T3FgCc4oxQti2JMkXEbvXk4Ny1tqANttDP4QmgV3A6a0eBKpK0b_K_B3a/s1600/vid04.png" style="background: rgba(0, 0, 0, 0); border-radius: 0px; border: 1px solid rgba(0, 0, 0, 0); box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><br />
<br style="background-color: white; color: #444444; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px;" />
<a href="https://youtu.be/hRJjGhoTQkw" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; text-decoration-line: none;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj3JkbnQYmcFNMi8Uowm2CsH1hDyx_qRc1OKh-EycVxrN5eatB3CamDxh7OgnNYjY7_nQsqVWQlmOxS8TcaKTStYweJg9gOOYkui-AE6JYkPsZFW-brlgov_xbx17UYjaEWPjFhfrjX_2U7/s1600/vid05.png" style="background: rgba(0, 0, 0, 0); border-radius: 0px; border: 1px solid rgba(0, 0, 0, 0); box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><br />
<br style="background-color: white; color: #444444; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px;" />
<a href="https://youtu.be/YM6ClYNfyks" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; text-decoration-line: none;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjP_SIzbaKFuCELKaHxIs6RTWI_zgzWtc9TPk0iL6g_Ty-P4WukZ0IPx1OIHSSH4IVQyHfEXAjzf3fSgjYqzTnHenc9Sw7XHdsMCkjcGVOvERzWHNCI1bdgfC3apJ4cnoSIkVyvvqlEVE1z/s1600/vid06.png" style="background: rgba(0, 0, 0, 0); border-radius: 0px; border: 1px solid rgba(0, 0, 0, 0); box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><br />
<br />
Absolute Value Inequalities<br />
<a href="https://youtu.be/bFWibZEDHBg" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; text-decoration-line: none;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPnHvDD2lxMESsxIxT6oktI-pL30HRZy3Iv_S7OUmCcJ0e6X4NE2FNGlJox1AWHbQbXpQxS3m_pgeAcC4QstF0Z6Lk9xvmpAVdFSOotXtqlF-rOUks67qjPNAdqoXFl7wVFqX6i5RYEP-a/s1600/vid01.png" style="background: rgba(0, 0, 0, 0); border-radius: 0px; border: 1px solid rgba(0, 0, 0, 0); box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><br />
<br style="background-color: white; color: #444444; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px;" />
<a href="https://youtu.be/3V_i9mT0X2U" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; text-decoration-line: none;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiv1UxU1aKqROAH2PCj_snv2pPFTAg7e_2QEkLzLVPuxytTCFmnJ8z7FwaSxuvwJ9Fx7BpizZ1Ng2sm7lphuIuK6JokYqF2b_VN1BSavl_XhPMpTLI9O0qR1j6IGV6QU_cvYZFoTNDRusbN/s1600/vid02.png" style="background: rgba(0, 0, 0, 0); border-radius: 0px; border: 1px solid rgba(0, 0, 0, 0); box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><br />
<br style="background-color: white; color: #444444; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px;" />
<a href="https://youtu.be/iF7g__2x0LQ" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; text-decoration-line: none;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgdQIck6A2KK_nQwvuVOuTfzNptCVEeBjFGSf2Mw50aSY9BWhNW_jh9dKEuZLvquDjPGCKpzYpIo-h4tV42hFUzXDlN881cPv2cFLLL7m0Y9u0SeXRay_jYTxrqW2MSapooXGKxdCLw_1k3/s1600/vid03.png" style="background: rgba(0, 0, 0, 0); border-radius: 0px; border: 1px solid rgba(0, 0, 0, 0); box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><br />
<br style="background-color: white; color: #444444; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px;" />
<a href="https://youtu.be/jOj5UQ31GPw" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; text-decoration-line: none;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiRMq6DzReqhDu5rVcYwL-Tf7xU0u5AjtzrvIY-czCE2fu7TF7CYqSPOFX7FPK9OE1n4vJUzIMeMyEyJ1wvTGtaEdH5Fk_4Pfm9cIpi3kQjgLGdxy0dvKIPn8EraXbC1_EGxuWycO-G1Rys/s1600/vid04.png" style="background: rgba(0, 0, 0, 0); border-radius: 0px; border: 1px solid rgba(0, 0, 0, 0); box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><br />
<br style="background-color: white; color: #444444; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px;" />
<a href="https://youtu.be/yA3NflM91EA" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; text-decoration-line: none;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjJpsLYu2Tf7A13Iv2B6XnxCVBL_-lUOeaCBBHIgGAZS7WR61Qz9v4UOqQ1yIhGw3cAQdIv2Ut2TIiodV8M7c4gDvd27N4JNSNQeCY3w_XLS_kgrL2kvVrVJi0XF7NqD7RM6PD6oDepRFUB/s1600/vid05.png" style="background: rgba(0, 0, 0, 0); border-radius: 0px; border: 1px solid rgba(0, 0, 0, 0); box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><br />
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<a href="https://youtu.be/y4vUak0LF7Y" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; text-decoration-line: none;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDMsvq_1YopTa712VWqoTBxFVFexVUM-79KR3bpwSkNPZm-MKmEL-OSlZ2svIs5419k8gfZaRqE3X-N4ZsOqjBze-G7olkSlfUlMKJ9gU5jY91veShrSnFgZU4nq5-hZnM8B3kIV2sE_fN/s1600/vid06.png" style="background: rgba(0, 0, 0, 0); border-radius: 0px; border: 1px solid rgba(0, 0, 0, 0); box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><br />
<br />
Quadratic Inequalities:<br />
<a href="https://youtu.be/TjNVJWfbWnY" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; text-decoration-line: none;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgw4ynSjyIaEr7YP2rw_wOpypvpUwtKMgACI-_08f4uOfV4ivZ7jnhN8H5GQDsl_ls5uksySaDbug24z5dcbMeXsFAEFYR1jvFOFe8b2SDqwJi_M-Cu1TDSpSX9SqwnG3EUusw4ZaYVV4Wk/s1600/vid01.png" style="background: rgba(0, 0, 0, 0); border-radius: 0px; border: 1px solid rgba(0, 0, 0, 0); box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><br />
<br style="background-color: white; color: #444444; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px;" />
<a href="https://youtu.be/FmVCBddNyuE" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; text-decoration-line: none;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEifzX_Y-i4xLGGZPPQsR15UYzkRQkIre5EFAB91A0hStJF1rCQtKx0DfsAnugVai7iKPXgx2FxTxx0AmPhOwuZ4oeTEoCVWUmDzl3Fk1RGbejEiF0va_2b4pfUTB9qzIotnw69r9z8qwKyC/s1600/vid02.png" style="background: rgba(0, 0, 0, 0); border-radius: 0px; border: 1px solid rgba(0, 0, 0, 0); box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><br />
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<a href="https://youtu.be/2H5_Hq4y0xk" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; text-decoration-line: none;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjyx2dJHui2UBx_z0Eyk0h0KruDl8BXUZXSGH8RYIDlalwO8fLD3ELK285kuCJCDywJP9pwQIjrZRCUVVqCRWMHfzIFSuU42abBQfr485Tl3VKSN4AD3_IYLQ1wzseuN8e5mjTDLmV5-oIc/s1600/vid03.png" style="background: rgba(0, 0, 0, 0); border-radius: 0px; border: 1px solid rgba(0, 0, 0, 0); box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><br />
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<a href="https://youtu.be/LeQbD2j-e48" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; text-decoration-line: none;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcI8XdaUD4o1VAeRmMmiix9W2TMgbSV-1qqPfmA5WRRF9o3rTwSHHAE3yJnwl9rkYlm1SArXN_JXvcVT4c_Qo9s4ErvUxPihViN2yd9-jNd-_lW3K-htzT0hKwv-ig9pIxLjvlfOLoqLGl/s1600/vid04.png" style="background: rgba(0, 0, 0, 0); border-radius: 0px; border: 1px solid rgba(0, 0, 0, 0); box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><br />
<br />
Polynomial and Rational Inequalities<br />
<a href="https://youtu.be/WZx0cVD6iRg" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; text-decoration-line: none;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjLmQNUxCogRNFfzSgnDLGMvE6IXSN8gNYEq84vZYEwNp8iLWDx2l6i1hXwv9p-ylGCSn-QodjapbINFB_mgKWhcoVGVzd03HlLMQ9buYNVezPJyAwXgG9moOvvjUytISea1IzC_3J96cUM/s1600/vid01.png" style="background: rgba(0, 0, 0, 0); border-radius: 0px; border: 1px solid rgba(0, 0, 0, 0); box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><br />
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<a href="https://youtu.be/r11z6jabuh8" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; text-decoration-line: none;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjsS4shIVHn4S_cADcl9RV6WTKvNFVXoKqblSgOY8BaZhjtyUAGjkfoA63W1YjmKq3so5Btl6r27cMhy8iJ5Ax6CKxHQdQf8i4ZPvAFLeteziFWkasmu552mHbo18ysCxq8kSsyL0b-muPg/s1600/vid02.png" style="background: rgba(0, 0, 0, 0); border-radius: 0px; border: 1px solid rgba(0, 0, 0, 0); box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><br />
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<a href="https://youtu.be/DWrGbQIsvc4" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; text-decoration-line: none;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEht338fZ7gSQ8NDsRq3qE_X6oYqA1RskUDUkVvGrYThz1YsUOCJZhdSCVdtRhMz0PzqfwqCt_hhkMVW5GIzyPThtVhXVfU6ERLloiI-aFvmThfSxJF1vdcxMi7_EZD4Dx3VCMGIbqHPMHE1/s1600/vid03.png" style="background: rgba(0, 0, 0, 0); border-radius: 0px; border: 1px solid rgba(0, 0, 0, 0); box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><br />
<br style="background-color: white; color: #444444; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px;" />
<a href="https://youtu.be/wXkheTuo6Qk" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; text-decoration-line: none;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjfS1obexza755bU-oulNYAWkAb-Yv8ckmF9F1iraBpnMXIhkLrAAV7VJzbgAGnag2-IgFGj64TXE4iNmGlds8UgMItx8o60nfaXU3czx4z82nc6K95Fea5xMt-HZ36M7pCAfwxASk7bVIu/s1600/vid04.png" style="background: rgba(0, 0, 0, 0); border-radius: 0px; border: 1px solid rgba(0, 0, 0, 0); box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><br />
<br style="background-color: white; color: #444444; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px;" />
<a href="https://youtu.be/_14pyXi3xg4" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; text-decoration-line: none;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhRF2Iyc9LlXm4FbLbiHSOmA_yp8ynNZrja6lPmd6B0U0YXkZxlZalOXmQewcD4WowHBrFs1AMgsPGJYEFRxTMv0rCmcYfT7rcJxBaA9v-Xwz1BzpdYIbWmrvPzSWSXRUMMkw6jYr1MbDBB/s1600/vid05.png" style="background: rgba(0, 0, 0, 0); border-radius: 0px; border: 1px solid rgba(0, 0, 0, 0); box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><br />
<br style="background-color: white; color: #444444; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px;" />
<a href="https://youtu.be/bmQmNO9EnZ4" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; text-decoration-line: none;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjn1PebO2hykIn4na1sojVUuAETnm-AswnEwagZQmWfC9bVD0syud4o3xYakwJ-ELKF6wCVnxEFTOSDFO8wvN3QLuqTeJuj_DvcFxb201OTBR9pDSSj5RSTmtMCF0VfOORkEwP2iOXIrNMx/s1600/vid06.png" style="background: rgba(0, 0, 0, 0); border-radius: 0px; border: 1px solid rgba(0, 0, 0, 0); box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><br />
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<div class="separator" style="clear: both; text-align: center;">
<img border="0" data-original-height="427" data-original-width="591" height="144" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgcJRhC5lm5P5B7Dk6iAXYDMTKcv2xkrf7B-ViR8w8Gb9NKbhVbP652a6qNQ4k1yMc8yeLN6XjaSf08Ph6ZgTDAP4sjwxsGGBD__Y9_FLTwKF0RqpcOmlRzXS0NNjsiM8-RSzV9QecyzIs/s200/conics.PNG" width="200" /></div>
<span style="color: blue;">Free online notes:</span><br />
<span style="background-color: white; color: #444444; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13px;"> 1. </span><a href="https://www.openalgebra.com/2013/06/distance-and-midpoint-formulas.html" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; text-decoration-line: none;">Distance and Midpoint Formulas</a><br />
<span style="background-color: white; color: #444444; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13px;"> 2. </span><a href="https://www.openalgebra.com/2013/04/circles.html" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; text-decoration-line: none;">Circles</a><br />
<span style="background-color: white; color: #444444; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13px;"> 3. </span><a href="https://www.openalgebra.com/2013/04/ellipses.html" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; text-decoration-line: none;">Ellipses</a><br />
<span style="background-color: white; color: #444444; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13px;"> 4. </span><a href="https://www.openalgebra.com/2013/05/hyperbolas.html" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; text-decoration-line: none;">Hyperbolas</a><br />
<span style="background-color: white; color: #444444; font-family: "arial" , "tahoma" , "helvetica" , "freesans" , sans-serif; font-size: 13px;"> 5. </span><a href="https://www.openalgebra.com/2013/07/solving-nonlinear-systems.html" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; text-decoration-line: none;">Solving Nonlinear Systems</a><br />
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<b><span style="color: red;">YouTube Tutorials:</span> </b> Click on a problem to see it worked out in YouTube.<br />
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Distance and Midpoint:<br />
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Circles:</div>
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Ellipses:</div>
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Hyperbolas:</div>
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Non-Linear Systems: </div>
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John Reddenhttp://www.blogger.com/profile/02231088988057526726noreply@blogger.comtag:blogger.com,1999:blog-221233850938098964.post-77111810530734045302018-11-20T16:49:00.001-08:002020-10-06T22:34:11.796-07:00Function CompositionHere is where we introduce a new operation - composition of functions. With this definition comes new notation.<br />
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<span style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjvtgvJOp4fYUQKyNGK12jsKsrpgncde7a4O20oGjogyv_vKqCpv8VPwbMI6sTouTOI8d6BLY1pbVuy75u537o6rSD_mpLd5MEEiDqiGkzN8L8qJddsN1PESs7R29bG06QAgMrhZSOvybQ/s1600/def01.PNG" /></span></div>
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<a href="https://www.blogger.com/Introduction%20to%20functions%20playlist%20on%20YouTube:%20https://www.youtube.com/playlist?list=PLmN1jmOiJEf5i_oL2O4CXVGFayy9yYsXo" target="_blank"><span style="font-size: large;">YouTube Playlist Introduction to Functions</span></a></div>
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This new notation reads “<i>f </i>composed with <i>g</i>.”<br />
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<span style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiLbdDgNwT-jVhYlSZp7BQ6ZdTLEWtXOjH-rOWMT01UB-ub7FJdJMX_aP_9hegE66AURjd9-z806jjDmsk7d-8JNYrJvoWGGeBoyKe3BxK-aleQstm6SOIYHGh96_nnEhliuQavjMhSqoM/s1600/def02.PNG" /></span></div><div class="separator" style="clear: both; text-align: center;"><span style="margin-left: 1em; margin-right: 1em;"><iframe allowfullscreen="" frameborder="0" height="270" src="https://www.youtube.com/embed/67ypRbrn2OM" width="480"></iframe></span></div><div class="separator" style="clear: both; text-align: center;"><span style="margin-left: 1em; margin-right: 1em;"><br /></span></div>
The idea is to substitute one function into another function.<br />
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<b>Given the functions <i>f</i>(<i>x</i>) = 5<i>x</i> – 4 and <i>g</i>(<i>x</i>) = 2<i>x</i> – 1</b>.<br />
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<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj7hnZ4OcC3Kg9PBptHAu4CIH9659pOx2JNP_KFvCeCzaKJk8QEFQoxJoyj-aP3T0kmN2vfilI9ZD5hOlrKscTVVRZh2deF_S8By5hLbsKGOPicSXXD2DBH4lHt_WtpdLpF3A4XfXWDv7U/s1600/ex01.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEguOAuJMmeBq13MCu7Jl43AS4s8G1HHw1aayJ8EIsG6Yud9FgxbsXxlaTncqvgSkTUsYqol_mDj_TwfOM9uCG69NrGcpxA9YAbDnFrrPlSiy_z3L259pfkNg5IXeahzOk1V6_iwT5YtnaY/s1600/ex02.PNG" /></div>
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<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9kUnMaVMaACsvdyYQ1CC1xYwnveV8O7oIEx0cc79QpIlDCMeiP9cvoG6yYfD63pDCDe2hLy63i47Td1bOmbceXHbw8Yova3V_wmKOVFcBT7JT5a70X4jj6DEhtvSdvH8hdqyoQ8M8ryo/s1600/ex03.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiv0eWBAaPKZ6Y6c01x4zGqn7Mk08OMmtPQ2BS3YJhQ8gSPSreTCBfgxekVZL_ZpgWu4kSBZWieMxEu7Dj3nZwHfgJEtwKXoBgsf1BnXHo4nJ4SDE_9Y8FBxpWezz2hrDOZLwKBJLYxon4/s1600/ex04.PNG" /></div>
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Composition of functions is not necessarily a commutative operation, in other words, order matters.<br />
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<b>Given the functions <i>f</i>(<i>x</i>) = <i>x</i>^2 – 9 and <i>g</i>(<i>x</i>) = <i>x</i> – 3</b>.</div>
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<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgKSRluWcvLWIvvRJAVl0lDqgWzEwVn-yt6SJ8Z27amlWHD36JYCsLemKd2QMpYAWKLCZzNxKt8jjrLPDwc1CBGOo00P9FpB4CrSvcYBJY3f1WvesFtyRHzpBwJ0bZdvHkWs1kBpTxQJro/s1600/ex05.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj5kBD_ZV-vb80d-MKes4WxfifcBZDQtr3HG718xS8LebOwbBiXa1JwTayTH5Dh6SIxqxRSN628m12-L8N5Sdh2AFZbu9W3jFns7P5eb2_tifn8OPAD4lrdK_JQ_a6Rc5ugFUnCaB9sqD4/s1600/ex06.PNG" /></div>
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<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgop1N00UBS2sGmRFPQVCO5bMjgSqzA9MikBwfADq9L_aufXFv3-V69m7Sh0Tn9ezGs6CNJu8aIQsresUrcG4FXTf0-pqkoPQsEqhZPvr5BhfBz0eeZj-lnPBXSPhnjX-Kw4I0q3znP4ow/s1600/ex07.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiuFmJZm-4Bfl1qpYa_szpWQJANqqmS2P-jJDkvuvFC4jCx3gzEWXv9zXJWwu8nS0-4uqnv9pNqepIRnm6jjVXPJstx_oI_pLI6ZQ9kMMB_7no1ckHzaOOEKgvaBah-mp2W1DjISwCWKOc/s1600/ex08.PNG" /></div>
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<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh7Diact-C3nht035Axi9mZEeUEoiRBzTsXm-aLKoPhUtG2YdLGd9fKfqQlUHhHptGh5d6G-kSKyneI3ZNIVvTl1Gj5erhcok01eTIpqkFTlYF_nDVlgkrSPmcJ88HK4bjrC99xin8Dw9I/s1600/ex09.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiFY4rVkXxvO8tpHt9SxuA9PJwIeHxfcT02mGYvgrlJQkk0GqjDxNTOtvCT6wRAB8b8TKRDCOqSzaCvw4cHhYjdXqFQjyycjxXo8cSXhS7QzCR2j8laFbmbInGYe-pxFLCbK0FzcYC6eNw/s1600/ex10.PNG" /></div>
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<b>Given the functions <i>f</i>(<i>x</i>) = <i>x</i>^2 + 1 and <i>g</i>(<i>x</i>) = sqrt(<i>x </i></b><b>– </b><b>1) where ( <i>x</i> >= 1 )</b>.</div>
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<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9ZhQaOFIBsAe6krG2ag7rDcobMYj7xAosmG86nV3NsDLXCUzwzB9vuVGSs6RYpDR7FWspdgnayZaYIktar6BLmhB2PZsbATChDwGMhwgRv6Gi4-PQvS8mZgN4QsXCMDG2eoO0Kjnmt6o/s1600/ex11.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjBZtGIVRopYNLdrv4Amxczu0DuSdGXxZhgdu3quusAQw2aJ1zYcFrJJ8VSnEGA0qE9E0b57dxnuwyGrPgqFpAinLLWr9iSWOMu0ijgsJwkUKcy-XJjxr8nFz3SZGWSAECKVvEF_0HwSYw/s1600/ex12.PNG" /></div>
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<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZg0iKdVrgkeCug192p4kxnzG1XgXlqHTdY0ifJlt5lXkFEgJxZKaOtNB5A7kKV0T_r_GNUM98s9-p9H6quxotUfFMPpKB4ZvQLn8HTQC1wHlWJrC11JjT_2GTljWpjcCgrmrcpKlBwMk/s1600/ex13.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJG18hIz8bIzrji4Sr-0S4ib6nNhk6czQCTx3O0sJJAmll5hSpQmxMgdw8Dyb87YkkJtBDdaiXxUfOz1caEUTqei-8xMXO8IQ05R_pztkCx1Fw1stCDCmYtguUf27OBGr1D2DFmi6sJkE/s1600/ex14.PNG" /></div>
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At this point we must understand what happens to the domain of a composite function. In the above example it might appear that <i>f</i> o <i>g</i> has a domain of all real numbers. In fact, the domain is restricted to [1, inf) because that is the domain of <i>g</i>. The domain of <i>f</i> o <i>g </i>consists of all the values in the domain of <i>g</i> that are also in the domain of <i>f</i>.</div>
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<b>Given <i>f</i> and <i>g</i> find <i>f</i> o <i>g</i> and state its domain</b>.</div>
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<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjJ4TmJ_n5XD29Ku2URiQ_-dUwjIuIgRcLKD3_himHlwT5FRTJzCv-4sQIyhyphenhyphenvFiXtxJTmwZ0FNbymdyvE0m_UpNPrRS0zY7RKzA_Zb5YKZFApnzMMobpgp_9qWMceU-8PYoUFdF5TOfH0/s1600/ex16.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhrYiIT7STRdxJgNxXad2rl2AUpDAzdIF_MukMlyLp7n8BOZ4KQJvkmVMaR5rJCACRA8qg-rKOcpq41Of1Y1mkHTqZwITbEsGdFV4zV43O0PaAiEDnmy8TNJYmPuaU-RyhA72hEaadv4Qo/s1600/ex17.PNG" /></div>
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<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQDVOAYgex3lLGWcfmEa_SnSyXjxODgqkLYf3F4yNYO-D0sYpsJauqMYhyvGyiLHy3IAMWRPnxrDb9QVJlxiRSuzNxsmel2zmjrAbgDkX4RdRsmCBDi0XBO17Bi3NXtUme3pHiP_7NbP8/s1600/ex18.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiEqP-ubhdvmngjuSLieSWbpc5QFA0dU3fy9Ies8d6HhJOHyQa9U43e-Fw1ynUj1AWENNIvZc5rgnkKyWsxf1AUIG4ooEv-qgDXs_HKXBBaFQE8UqhNlLIfr8OwqL6xd8i-nXPUSAMhmxE/s1600/ex19.PNG" /></div>
<div>
<b><br /></b>
<span style="color: red;"><b>YouTube Videos</b>:</span><br />
<a href="https://youtu.be/BPUmKB_46bQ" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDisLrv2GSRXHuYbKUUDfryQ-49ffSI25cGiU_w2_QlDuL6wTgGdtM3AGj42PLuxjb3Tm2wSEzu9hN5V1HqtzRnECVxWw_PRQ3jv10zzyDJeAxhEql3SkCC6B1NCakruLttSN_qFbAkOY/s1600/vid01.png" /></a><br />
<br />
<a href="https://youtu.be/fQgdbl-IUPI" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgcQl_IyqIctlAHAl3PUgEZkhcfEqzNTYegLAJZKg83h-ip5LTk20rWEQUWE4s2a4nGNs-FwmW3w_SOh72VODmAXgS9JfowAU6YTnw7FPbnBGIX6mVjh9HR6ORN1rrpDYK0CLtPerPRtHk/s1600/vid02.png" /></a><br />
<br />
<a href="https://youtu.be/Bc0t_xovRTQ" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjA1P0UJvKXCcw4L4DBklV4rwweYuiAtfzJkN7kQPoQVFp3BLqoZehV2PW6jJ51pudsUhPYC02ju2rJiXJ2NgYcMpxrEoISHq2dSKJTl5zmvi-q8ejZ-NG3G8G-8eUFdw3oczKQU6f330c/s1600/vid03.png" /></a><br />
<br />
<a href="https://youtu.be/BPvZZru-11U" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_Lqg9zRL-Yc-Aa1IrN-h0gaz9Snz9HR9eF0VvoMuY6gs0Jd2D7HNmKDLjXeNk8XC2JtBCtr-2r29Vs64WLl3HVT4GLtd14yblgN4nhQ_o0DxAlwhHaMF0fETfe1J2rWk58nKZpJ-ijJA/s1600/vid04.png" /></a><br />
<br />
<a href="https://youtu.be/Hct0vSqV4PQ" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjx5PBwJF_jSGQ10yU2X-RWowgIaWihvhzXjolAB6dOvzFxP1PsSGmBFN8NIvX7zvh1r30hj7aYcABUvfhAwIuhsEYG2nNglnCw6WSYBxKh_nXyk-GKISa-lB0cV8iG2aJf0T61Q0vfXFA/s1600/vid05.png" /></a></div>
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<br />
<div style="text-align: center;">
<a href="https://www.youtube.com/playlist?list=PLmN1jmOiJEf4hOOt9DsjxGV99vBDSMsgk" target="_blank"><span style="font-size: large;">Playlist on Radical Expressions and Equations</span></a></div>
<br />
Use caution when solving radical equations because the following steps may lead to extraneous solutions, solutions that do not solve the original equation.<br />
<br />
<b>Solve: </b><br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPrJb4n0IdkG8rjzqQJ5z8Du7Saar-Z18BxZzeMREMaw0fqL23t1dbV2B6pE8NIZk3kWFQL7IegzaVlAcpZSZ6s1pTHXk9xPWVqYzJO5vH5YRUyaLFQy8aCs5RK_piN0NU1kawjBKCQV8/s1600/ex08-0601a.PNG" /></div>
<b>Step 1:</b> Isolate the radical.<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjOai0LcalWA__AQ4eORz9HQbrx6cIrsBLkXY88GF1AgCB-bYyGsfG4AoDImSvdZGgI-Ur-yZrjnN8wSEjt9oFu4wumqKpi6a4nb2OR9mv6eh4p_8VZm5j77dC0uP2GzoxBPvW3CrkWTtM/s1600/ex08-0601b.PNG" /></div>
<b>Step 2:</b> Square both sides of the equation.<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj3MGMJIbb1BVTeCTN38AlBBu99yHFSHS1E4NFB-2QKMPAN0uTObQI37mk1NeJ_0rwdgxhyphenhyphenP0S3k9I4EBD02QPXiM05Cei70RaTDGzeYzRQr6A0A6IeZaHxEUFxGgG97zx52CMZhmLjwPk/s1600/ex08-0601c.PNG" /></div>
<b>Step 3: </b>Solve the resulting equation and then check your answers.<br />
<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgPbg6XKrOVZQda7InFargflRhc-qOJ1TrUtO93KgTWU0Y-VsclYks_9qRTyWR3ZZw01MjVmr9KJ873jJW-Y_z0jNDLp0P5QHRqly_w-phthuLMGMxoQ3CWuNPcGKTtD7C2o4dmkMoi27U/s1600/ex08-0601d.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi5rNH_KB5qg9r9K67PnPadWI2BuJXzF4iQrQ7KftlM4MBL0bEn3Z9KhD5LjTtNz3noFIe9J5aH5pmlF4RIb3n0H1mObxDxGeJYdEmZZEtv8pF9YDbPH2dmaqd8RpQGkh6UEBVYqI3f0k8/s1600/ex08-0601e.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEikXuV0ctwFTGPXY56uU2GPOcm9zDcv4tFt_pimuR-UFyur-DT9oSvp0G1gn1RKl_hxuyeZJJYQH-GKlw3VP_nn3TCAP-r1T4YH2-1sx4gVfNfT6H2NkTaFfmuzacXLfTPvsj00hYsvi-c/s1600/ex08-0601f.PNG" /></div>
<br />
<div style="text-align: left;">
Whenever you raise both sides of an equation to an even power, you introduce the possibility of extraneous solutions so the check is essential here.<br />
</div>
<b>Solve.</b><br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYAivThPejUUmMw_mYylzVpiBWvWZTHTkQ-IjGEyENtNh-ll-VKWUaUmpK3UeeUhS3IThAvgxmJI8G5stYN-i82FxktIs22RoaZl9NgH9NKReaERp1DDGzSiwbGJt9p8JmIJQnzVb6gvw/s1600/ex08-0602.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj5yv4cJimHthHIx1r1RaUie97HrS732ueqLKprec97H0LpHrZ9lqtc504RFTme5ma40866n8ptF8PUIxH0IzrHqQLg55XKvPVQzCoFTxuL-EPlXgGrNlgbDjPyHQNbhWJjmg0efv5WPVQ/s1600/ex08-0603.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEij38b6sVOQ-ZQUd0ALkoFyevH-pqgZy9COrScI1QjZa5NFQbZ-8hZVxYzDLhz7HKh-lAd1KXPM35arPs2wi1JHbOy_QPO8HTjLZBwm00qLhzrWYq1UT32C5peTb7r77cICIAyGVY8VY00/s1600/ex08-0604.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgU2FXz0Kj79EObvroTt_OxW_W0quRoOYY_VDlfgAw7K74jqlqqrR7uRqOWQfDdbWwnKbugP8XooRczgAKwXxCD8tqQF40Phe0mbRezrkdhmwI6iASviVxd5otueyODHVlHiz_17XaRHps/s1600/ex08-0605.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgslaSPSKtVfWSfQjWvPCO60SrwAcFEvtEUTPfDYrxNSJIVTbmi8hjXUxrLzvA-4YNQ41YNPPfbFjZ6GuwqYuaTiXg1zadecC9SpI2MkIbIypcvcYn9Q_di7PlrWs1pccQDkGC5qFSKdJA/s1600/ex08-0606.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgKgy2G96wkar36fUb6TFpy5Fj8XSLL47sRY5I-HbphIlj5c8nvCiZQOTGqIFubLtV91ny3x_HfciY76CuH4RXVN-V0nrOqC1WFgjqLCMKNwkvskIHzHoR6JOraSZ6pssMiAMEhi_leBAA/s1600/ex08-0607.PNG" /></div>
The index determines the power to which we raise both sides. For example, if we have a cube root we will raise both sides to the 3rd power. The property that we are using is<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbdvjHuGZAExr2BICsn37bv6crJGw48g8ZMGIoPR9CeyZ1wYHspLpzMc4uuu46fxBZfidxE63aONCszjpX_gseLtBJV_Eox6dNHdYwDK433FZieJLNIkK1qUvUZnpTc0jJruFByiVvbPg/s1600/ex08-0608.PNG" /></div>
for integers <i>n</i> > 1 and positive real numbers <i>x</i>. After eliminating the radical, we will most likely be left with either a linear or a quadratic equation to solve.<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi8HV3Xx43CqMB_-3YVwgG_fz3vz3gBqgCEZL6oN7GZ04A-FPT9DD2ShBcaY4UrEUIDjE4Y0dKJRkHiW3VlauLcDt7NGAlFNbszA_dwrHi-k7Q3kCu04ev59ZTLhF6oO2lUPx7wZFFRskg/s1600/ex08-0609.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg7Uc4Utj6nCk5WRjuMyugihMuNi2m6GPwaw5fzAhrI7AU8fnnX2-ibjyRXNYY97IHCnb1S6XvNkgQlrq7CK64k6bUJMnzJhnUrDy5TaxbcaNwxtaXZqNfg4w1-vmfBDnAYSgyffDsyZfE/s1600/ex08-0610.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjJEaB2sywu6hhEJ9SwBlv0VL1iidemP7GRsPvO8X8E0udyjx_Gszml7MAausuY5coSCHmDtqHBB-PvkcjnVEHHPQEEPuaGHBYhrMCpBU0pUNPkN7v3JFIwGJuoXAdI9K5Mgxe17gY0yHQ/s1600/ex08-0611.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZz5FhbAc52rTJ8x5LNIMvDiEHgyQzIGA68bWoIPyDM-klfeIw0PlDpopHb3Wu9mWc9GNQbJr_DMtC1vOwxIrjKoHWW8gLBmfr3XWbUdWVXwpgJ14ZxWJlTw6kID9_o6lSpRaiFuI7rlo/s1600/ex08-0612.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg29WJEmBdRfxNabhrsf_dUCPYZOr57IWmeIBkuxytI3CFw0TUJ8vQjCKZitpNNqpULQq3o_jX4apIS8KcrxPPOzQSMV3SFX_dfZAuhvjJ30P8Bq_8Yi6bJj58ZbTVgPnQzsmtnGNttfw8/s1600/ex08-0613.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiBDpueoN5jUByLKNbfZneaQ2ncZRaUb5-AyAgiq4cTao0-QQxbGSsz8Tp8_O81kxqyqIddywFdErvz58ogWEfovHW74EpB-Wj2peCHxdI9Grcml_GpK69QlGSolP8u5pPxxsilebAZyvI/s1600/ex08-0614.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjtY0Yay1UjlntitrTlFNjAYWhmmZVPI5Dq7NGhNpuzxiafr_fZ6eMc-5-g0wNZHezlyWd85hTVzTCxOxE6av2HC_wL7iWEjr0WpQZ_3lEbvQbJpksXj6PaO7AGdQJnpUdhfOVztw14Vh4/s1600/ex08-0615.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi3T39BTl1D5WViHHsetJFSflS3ON20sd2mhBWHRzK8yjYqWX-y_ufpT_nmNYpsTubm1ivm9yQjV12cVW_BYr15N3Oh_uFQtiBBLum926foRm6R6oYfEFWHhsCaLk-zDzqeOpZDc4uufeY/s1600/ex08-0616.PNG" /></div>
The check mark indicates that we have actually checked that the value is a solution to the equation, do not dismiss this step, it is essential.<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDv-g_WKE8s7_TZsulvWMpXcXLNhJxJyiExzD9G77RHFizoFjUplMx4MscFg8Y7ZQgA1BV2Dxmsm9HvHWYvKXjC_60pEBA8Qm3S1SHq3JSxFzN0izy2TYb_AabGoPA1cKTyudwaExOgJ4/s320/ex08-0617.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjoQNamWoNrtPnZ4wlZ9JISQW1zuds8NhgF-B9Fu_hJkUh0tkPweYv3XyI3OK1f1RKKEbOitEev5VE7Eyxiy87ksOmccZNwK9cEC48I2Y0VA0U7SnT4z678J_E9oHI3N8rLOb54Y4HoJL4/s320/ex08-0618.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjwLbVxSk3mvKJ1sW-7M2yb-WFTYeKlxgqnYaZ2tKwEhhuxejWZwj6tpdtl6DjqhXmoPbuVxQGvGjqe8LTytN5ao31EZ8wOv9W7Ii2KRfRWAM72SjgkJDAyPBm8GWjRj5IdKTvZhIlY5FM/s1600/ex08-0619.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEitetLbN2qjInateQJw1zX7hXjIR34CD0Lk4oFSi9JUB3XFSUqQercp7LyyyahUlJg2T-VED8thX2ZXNvdxywMxVoGTbm7bZVqI1EHFETtOMAQndojqmGMECp9hHvr4WMT4z9QrAScoH70/s1600/ex08-0620.PNG" /></div>
Some radical equations have more than one radical expression. These require us to isolate each remaining radical expression and raise both sides to the nth power until they are all eliminated. Be patient with these, go slow and avoid short cuts.<br />
<b>Solve: </b><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjUU2RCCLQjjU6bghjcT1L8JPBxhYyZVmP5Ef_HgQsQZOWw-p4QY3YtudgcU7YiSovNXpFA1x6fXhaOLwCKMvFRC0h_1RhWTkltdbYU7yNL8JEJisoZTPpI8n5ewUz-3OeZFrJkQaA4Keg/s1600/ex08-0621a.PNG" /><br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhkbJRgfwNh2GURpMPuynG5lJiyKZ_XfZuKTVFgnQwC-9nZ4T3edH17kXbsbGTSHLxmCB3stgswrGLpOOmvzjzz5pLVjAibzBryBN0V0Pgu4KyayHkVnpsDWtNLzVIMRQvIAlckInN8arM/s1600/ex08-0621b.png" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgjW7oNr7Xg-LeNWhdC7dxNorLB92rj2wQiTS9amxrFW0w03EnfSIb98pU7pUaCnnfsCMyQelLvMkqU9ySKZMgJ5Qi3h-9uydCKs0dK4QMcwSy7hIvk8dwJv8G46cwfWVR50tS8HLOwlj4/s1600/ex08-0621c.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjD1a1SonwUbuhoZKmdhN4DV4bWsoRFEmzRQbZWeuh-ZqBws98KdLxQwEwjuI78bt2QboZVuQcm0GraozR94z4iW1_mrWI8lwpTgE_wtg4pRIxvKUqD00uEsE8hm4jc1Il9kQaGrUU3XlE/s1600/ex08-0621d.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEimNRW8vPflPMgvjQxL8VswCyFUU85aUBEJhCB28SNatZJNoQHw7FgUK011cpR82xVOr89B_yDEzlggZcFuMNseLmMgIOFTTLRk7Bsf9K1_FFLv1f_LcQnjZib555OLZkR_rfyAJI2AJ7s/s1600/ex08-0621e.PNG" /></div>
<br />
<b>Solve: </b><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiy57Pqae6Jz47TcVlwHjLIoyL1G5rbZab1ZpswsndW1FabWkkmMPtihN6yTGsxt8pO3Ev2r56Rr3JhtBxKmzAlHmC1Qo7u__2DF6xwLQH_vPK2FXqsTsiEhfi-ZudFageHuXolhn6VoJc/s1600/ex08-0622a.PNG" /><br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhqCWZKOOWprbrgzlaPUhxrtqQ_Vk9uFE0GgF9409abioibSwE53-wUj0wnaz-vJ4BtzaIV9c6hIGlsQQvSu8coux_83xZ5JZhVUhVsU78jJWkvixGZFEyVi6wfeeOx-V1nIj9rReVEoJE/s1600/ex08-0622b.png" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgFPzYvUulsqn3XLMCWUo1Wl0nWw3h1sgHdspEtwb8Jvbjl1QvXhexhEjZg74V289PpF3-VXPifHIJQaLfHdQrNjUtfxqsRwqiSt3pElV3haKqzsse9xi4q2xBTcrjHTK10Bl6Sn_MhX9A/s1600/ex08-0622c.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEizU-jWhfS87jfi8JbHCW2vwFQGJRXeaXWzz_MJV7ZwmnQ14ZnEdaFpL1Y67gdyaK0Xponw9hdqzYmhP275MEsexxXsg4KIVAzjQ8Bjkr6jFBNCeB5o69MYUAb_9l7upn8Ascq-PYrXVj4/s1600/ex08-0622d.PNG" /></div>
<br />
<b>Solve: </b><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgjdvbUj2OsYV9HtQd4HlwtYryrTmPD2yQXYDST21I8WAYlg1O3e0f2t9di3xQ1fptPXekMUGkZMxsY0Bb4px5RKsy9L1Ne0wF9AgAI-7xWzh9Q4M8XfpCuJCK2btLRMu2YNvJ_u1c4TfY/s1600/ex08-0623a.PNG" /><br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjEPbsM3HzPsT8dUSLZp4qDyqhV2bbCQKe8F7KFpJnTK3Z3W-qnhK70oahCDL1utdt6U6aozh7hJdqxBg72nGOCjm3k0Hkgc3f4KdWDH16nUjW4Ljl6kgWLiAngoL07845GE6pPAlv5Sq8/s1600/ex08-0623b.png" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEil_8vG3ujBDR9J7PUI9-DIgPq_QTrVoSaf7-s5-PjcOJ_zMhmy6_3i_XVdQMZEgAbhhvW7Wst5b1FhyphenhyphenrP7qIsqM7jdmZPzg_RCrW5CXCmLRwor8sXOGNYqbP5_c4OW0jaK5uO5j5ogEoI/s1600/ex08-0623c.PNG" /></div>
<br />
<b><span style="color: red;">Video Examples on YouTube:</span></b><br />
<a href="https://youtu.be/zlMHq08Adgk" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgjQbVKfTFqhXDH78TcCCMVfTPIK7qk6e4FD99H_zv9LTqxdsbcADZ8EojeIhw5zyIYIeN4jBVpNqs9MsCvWAgMaW81_KuYuENXppBvhA4LmiyVZROPS_vlPH6bZ6RO7vrvCw0Y1J_5jeA/s1600/vid-0835.png" /></a><br />
<a href="https://youtu.be/bY6DT-X9hTg" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhPjh2orVx9I1CnJYS45tIKnJ5b7ANT-z_UeVAeyP9nQnz_9b3qJf2YizEQOwn8UANHwGtnWZyfvq53O4_mZcaZtMTkKf0bWQsM1UovvQd05P35WoRAdkOSrPxJUm7yHaweZNzqWwwF1Us/s1600/vid-0836.png" /></a><br />
<a href="https://youtu.be/cneOXvx-n4I" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgQdWrM9LhX-FOxvGBcPSs1dFOSRPPFPSZCUVDDME0uqF8Rzdemaeci0tyBty5fakvSH6FZl1M3QHi9pZwwLvpbKIbyvqCWGLPhnjwBD9MICNHnuaU3ZK8hiNCV3cCtZTILQw961iIl5ak/s1600/vid-0837.png" /></a><br />
<a href="https://youtu.be/9icCm1p2clw" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj7L1gWbetk7lcvbpJmNJr6PUbHJ2jUgluszajtyjBcBycdrrmwE38gk3lvqCAH6qU2lfEc8mpaSNxbk0oGKco2vtOWEvjriTVCa6dR6hZFLxDF3rRpH2xQmhyLXVy0ODNk4-eaMAiDxdA/s1600/vid-0838.png" /></a><br />
<a href="https://youtu.be/O87kFST_KtY" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEizqpJTeIIs3qfYfIwdXzYVi2W03GkD-AjNj_3xbfm0h9RA-87xmBYMA6To_4xTWFpANu0g3qRPHdK4Ct-hHDWWENROtiuKruhhFIxWIm6MRyumniVN1aZp2oJGn06dM-NYqo6NGtvdMlc/s1600/vid-0839.png" /></a><br />
<a href="https://youtu.be/0tYFLH2cziY" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgCxajmMI70E-L83teoFf7EVwj287Vh1R4spWngzeiVC78GyNWNXx5ZMTRdA9ZgzW2CgP3zwh6ylq2eUGU0w7nY6A8_6vsoUT8A6RdqkV8J_WXKf-VCAFKSEWbe9ChKtXmkstmKoG33zBg/s1600/vid-0840.png" /></a><br />
<a href="https://youtu.be/kJMALAnb_I8" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgo3nNZ54kFst0cYj0HfxAGlY8mR9VK8voQWwE6iW0QW4sl-Ql6Fex6E41LI253ObOAM4ShcRr2luiTHNcRxs4lhaIVg8gY5miByDsf4ezAGemIEylsLdyLeSBuR7VvbSYeoWviyOagVdU/s1600/vid-0841.png" /></a><br />
<a href="https://youtu.be/4AmAZNCrEn8" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEheFJB5DZx2iCutjtvkri-_RJhdy2xWwnQ28ZROPB3syfkefEDVJv0G59fLnPq8-31n2KoYdUWXNIqi2ZB1qZrjC3KBDVLxyf9chyphenhyphen7VFcX5oI6p1iANzGOt5aGHhW5IDRkVxTkN5LnbQgw/s1600/vid-0842.png" /></a><br />
<a href="https://youtu.be/3nuwmnNaHdk" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhqEaaj9LC4AXnZxyqtwLokeMzRQcCTc0JzypBPrmFszjpsEkIQdSzEeF2mS22Xjlq3Q0TDcRhuLJ07LQ-WVeZFq3JgYSesHG_bMxPkMJ1ERkR0z4SyAmHD7gCKKhRgbh2HdUO1oO0nI9I/s1600/vid-0843.png" /></a><br />
<a href="https://youtu.be/p5OAU6EscxY" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj8qe_mG0dJiA35RJcVkYLca866fS54xarOHoqpWVXKxrKFLhsxInpfY1iwM2VGzewwid9rdH0GP3xit8pO8_sVtQQFuGpLBu3IkG2Nfv5HRR38gLU_P7fYPMmJLuNkyu7lePu09wJstTg/s1600/vid-0844.png" /></a>Unknownnoreply@blogger.comtag:blogger.com,1999:blog-221233850938098964.post-8832825983745519462018-11-20T13:59:00.000-08:002019-09-15T22:14:31.802-07:00Rational ExponentsRational (fractional) exponents, indicate that we have a radical expression.<br />
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<div style="text-align: center;">
<a href="https://www.youtube.com/playlist?list=PLmN1jmOiJEf4GszaRKkFFPeOZgVQ_jyjB" target="_blank"><span style="font-size: large;">YouTube Playlist on Fractional Exponents</span></a></div>
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They are intimidating at first but you will quickly get used to working with them. The denominator is the index and the numerator is the exponent.<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZXErR6qo0lBI9ut77BFfcWQGuq6kkxAN-f3IHzKlhgLcldQBIzGLyDO3lJO5Z2ygtwI-EwvrLATJKZKmcPyArnmKwpje4FxsmbEe2gZfNylcMpIbOGcrSpyc8AJCq6zhJz5E6xQiLo3A/s1600/ex08-0501.PNG" /><br />
<br />
[<b> Video Instruction</b>: <a href="https://youtu.be/LLeIOpcpgWQ" target="_blank">Rational Exponents</a> ]<br />
<br /></div>
<b>Rewrite the radical expression using rational exponents.</b><br />
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<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgyQgWxdIuq3HEim2srVAjfIWnzx2w4QTBvtzyFcZgO-4EDqSIz5lbKpReJL93rfT8h_9sav9KpK5Ej69FoBop1H25GfrhsFK3oHL4A93pHNDsF7cEmktTmpri09ymazkaAITmMvq-hMjw/s1600/ex08-0502.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgzSwTa9xrjTL2r-JwOsVfPDqon5H8ryKRPgjrbX2GCfCTYhGaMKcm3yzCllC178VnYkvfhQ8zfaHKU5lbQb8LDXXTqMk4qKZZtiHjV_wuUiqLYfM7jM1r6zYlk-HJDtzRUvtBfrCTjEFQ/s1600/ex08-0503.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEghRUDfFxSba_MFCXV2V4bud-XdfD4ORwaL7bfTm1e5BdzufxCvdaOiU7Upq39HH7Q5ZdYRQR9mN3Ct3VCFwAcke4CO9R4S1uDmBVRWtI3gxpsYVzwADfPaO-BfXTsVpmQHSsvCxcCUq5s/s1600/ex08-0504.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6IaCt0_s5PszAuCgw1BLxqqyoJbEXH-dnqqO-vUWQ4E8Xt_IB__KVC_O0PrsKZHc62WkIjA5wPHtFRxxZIQ-Dz3DlWjYnJgVnkdU8rsu4RqvVbbZGYWwjRpesxuHO_FskLvy_bVBoDg0/s1600/ex08-0505.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxvpb97jzlVqG6vSK-7J8R28V-i4DsLYKaLEEDWlXZSLLPft-dbb6TOh78oMT_Bva937foIxw3VlPH57VhyphenhyphenCSRMFXpnqjsGFF34Zss8kcCQYin7cWg_0fkEKqljpxW8BsBTKKdfXiJJD8/s1600/ex08-0506.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjyNndWMJm9qEAAklFVlLJdXl7VymtqesoI-5DBSGBO2WoeMtzTsAEPgYmh4W4QEl7c0aYayotdL2t4ZOIZen9dp30maKYz42N5uw8amkEpFBnhyYy3z0SVTtoHDO5WDGQzKT9fWoj_KcM/s1600/ex08-0507.PNG" /></div>
<b>Convert to a radical and then simplify.</b><br />
<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjj8TLg5N4yYlr-1WOBzwYWiKnUzqtth6Bg-2o9Wha0noJacXAa7DdtVhg6u4012XaFT_4yrWRoT1R-iXmicf7SOOL9NfcmykQhv3nMinGt_FzE2kprbxdsxsCYi0kKUqAEPSS84IXooKM/s1600/ex08-0508.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUVVsuHkwE3NaUUvv87TrKfxMHjF5j86CMzogSwyv-mlj9f4CS5k1dpQ3ho291PNI_1lrlnPhG8H1XuGJxbyRc7-kLOm3N9TJ3X51hLaw92aFZ7RyR0ZKojMfcrolLHVwyOuAM5MYFhpE/s1600/ex08-0509.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjqgOJ7r2j6g0fth1H18x3j9yKE6hV4ck8Xa386AI9PlmDBcI6I9G-o-EfBKlOz8S8nm5wejtWu_cmrcIp1WBARMAi7tcX6sESndLcKE63DeCgxizErs3JHGkJ2My0avZIxfeG0KYdVbd4/s1600/ex08-0510.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgzoKFJcgdvX5ssgFIVZrLHTsYFMKoFpsJsOQcPfcxc9epT747bFLuY3UVVHzWNlShqKdN9ag6kNzicXmhMrMtJkmMFuNqZYnZK7QzNOOLwCWD4k_gF4zqvDjfiefAar01n4ICVaLPIaH8/s1600/ex08-0511.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg1UbqWbHcg_983IZnveIMXeX2SQyBoCrJxKsODAj2Gj0ts2C4FpKcCxvq69eR5o1ip1juhpiPWSrCKAMpP-GoFzARU8cDrSLU-4b1ACkQovGSxJXIi5Km1U9W54nyBxnfSPjsS1EeL2hk/s1600/ex08-0512.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhjk0SmODBzzx4HrHI2fBCyPXySznYHFDMsM6n7KVcenj0QhnvaERJ5f1oe3Sm8PoAWOI0NyhZ6a2fRwH23bAjbx9-Aq5eAyWemsKBXry_eeO8o3PX4YQcCDwjzlmDmwoC_DkPnbCUbDRY/s1600/ex08-0513.PNG" /></div>
When simplifying, you can use the rules of exponents to simplify or you can convert to a radical and then simplify. Both methods yield the same result.<br />
<br />
<b>Assume all variables are positive and use the rules of exponents to simplify.</b><br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9so6g2S8yonbDIC6Y4zGICxU4ahmkmdoRSXf2Sng0JnIq_CAXH1M9unMrwOiLG_cvqt0EMtvlhkao2gHyXWmBHeB9VVS4zlzGcLjcwKGEupPhbRo6rNlKj-WyFjbjAlUnnm1D_il0ulk/s1600/ex08-0514.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjAJqwznpldYq6yi572AAgj3zuVf1V7JsTieVLI_ye771KCWO1eEkCDF6UVRIpo9SMlJQfiunQwqdBsGyzFKcjy5PZnHd6o1FG3Wl50VR6-H7XZezd0QWSv6vln8ZzfkXKdZmzaRv_9UVU/s1600/ex08-0515.PNG" /></div>
<b>Simplify. <i>(Assume all variables are positive.)</i></b><br />
<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgUF-5kRBgW0hNvzeFiG6khmW-D4HdNG9h1yG9z354itNPdSmoIo8ACVVrBvxXAiNj5gv3f77cZ9AjwbaMYnQ0cdjRPHiMpVAdPclLZssf7OEPTviHFkxLRBTBoerac7KrDfmuexb0DX5U/s1600/ex08-0516.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQaAaeCH8v2frK0ORMWJpDGmm5BU0x6VC2RvsmoKh_JfBThuSOVOxYTfFCZm6jBCRI63VyYbVIM5nsXHldL_AXB-kZGeWDwiUAy0XFTizYceliJ0FrrDUfmjIJaJ19d8QEM9C0ie9hXM0/s1600/ex08-0517.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhICpl0ZLcz9jtOgURbvhtZeDYOGQn9AcosX-53mUmECLMJru7VYGc7IirMFxCCFuPPSV5MrTdK89u1RT67NPb7bkqhI3PB4O2gTmUcboWt0RW-e4tgwWsK5lc4j40WyRMv4y6wR4I1yMk/s1600/ex08-0518.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhq2NOpzx_b_4XIPV9Dj0S808cyzOFyL4R-Mazek_2ewTR-MFS9wSwYVUm4xiOYSbPmmXkpOmgvpPvdq98REJrk8pezEi3LOyN3bHWbul-cbo6rwDCCRmLfUp4H6_sGjf9MMqApreruu5s/s1600/ex08-0519.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhRFI7ZZg_OdXi_iIYyfxuKR78vq6eQ8_lv-wTKQnKFs_8RGh7P-lfS7qLQV_QUa3uVmFwGRDjLSv8UcH2oZxhJkNvcn1jl0-vqNiZLJ68_xjf0LL0KBB1ZMQ3dLjRqxuwVjmJO7toOBso/s1600/ex08-0520.PNG" /></div>
<br />
<div style="text-align: left;">
Notice that all of the above problems worked out nicely because the exponents were multiples of the root. This is not always the case as illustrated below.</div>
<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEirgl0nUmX_VKekHprQdG62Bi0CyUpwDHgbWUe09YXEeaIFdLRiFtqTMi34u-FOBxUYDmBvd96u_0SYZcjsZcPbXAjhz4wQNfRlt0SxlIdi0qMRjXlsBfUkE3-Cr8VIlLsCF8hw4tXCyX0/s1600/ex08-0521.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgv7nTN-ukVL3VL23ufT0yMnICl6e6rQYU_hk6iNj7rbsSMx7sC3guGpfi_O0eFZj0-mvSD8zw2x_YMWOkF9iKhqGG9HfFyW9BFGkkfD3UkJ2a7L5nxE112ELG8Ic-ZcIdvhhudRCBNdbE/s1600/ex08-0522.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiX-AiHAAVW7KR8DzK37q-8cRP1GiD2a7jySaFomjvHbWsiQrjwKeGngHZhLFFBCCRCEgMXukZKsP1YYknpScUbKBAg21PYcFe0fYdHcXRE5pOfG5EhyRMRnh0naMFUlnxYSXPXVw-chSc/s1600/ex08-0523.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-0eaZqHgEAJ2e62Mv4VUPdbwJ7o3PfV4UVmd7MVAKgrXxQFjgAAvBc_jvUASGqqs3gKKe4erz-q3Pcr_4qIGOhrh1vWPhS83yXX9mjSgQ-MrUbarq6VsY1bVSEI16IzWCgzpcSHV7CKE/s1600/ex08-0524.PNG" /></div>
<br />
A quick way to simplify radicals is to divide the index into the exponents. This will tell us what the exponent of the base should be outside the radical and the remainder will be the exponent of the base inside the radical. All the rules of exponents hold true when working with rational exponents.<br />
<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhv3lQX8JGrmAF2TxMOAxzV_wZsvnCEubAZs0ww8oV9jxbpGe5osPy2b-BOR6k1Caf5Dr3KjcnKcjcmtLCQmOXppzi1TWyFTwoQPsh-cT8JfGDMgb4BaQlhyphenhyphenIByk6IEgTUu1DAYO50mT7k/s1600/ex08-0525.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiV9aEiNpKhIp1WXygLwjCuNx39WpuXmSRZc_O_6M2Tioz82KMx1jGMb0McN-iBUOA5MeEdFFMX5tJ3xSGEB1vkkQyGu3CM0SQjine6ya_Yet92gqvt0Rl9sVAKg3SwEhB92fqhyphenhyphenjIIlTM/s1600/ex08-0526.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEim2ZOGBvC_k8__J2zvc16qEsbZ3sVNstAm3RVKUCz1lfvtBR31AlLBQCclUCMfHdiLxTSSYPq7XYEZk50RTg0ucA2KxDKvytg6uGjHOxdmo_o4Hb5uWu5QzI5tSCyttlphJ90g28JR5ik/s1600/ex08-0527.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjXUfJd9g1GzXCERHic-9xQnIcScFRC68mEXSki9HtrSCGJT8z61NgEbjJRhXIduXQ3sbHl86y7KOJi_Ytt7KGuz574h-YWYeyhb7JZC0ae1qOlbtabD_oGcjCiYfFmJSy-ehs3vtB30sY/s1600/ex08-0528.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh8RSwwxKlSUcP3OaAbdpX9ExLWDHmngz1rxKWTRadiTIx4DsCLFOxVSEfCKMfDc8wj89jxRx5SzumhYEi9POcWaVTB9hoZ3f_ZIApftgZ7CQlrCGr-edEyzxgO5F-xHEh5uIM_yr_2_lQ/s1600/ex08-0529.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhTwUbe1POoOt7wSNtBpwCTtPm02wFosSvjw2xTlRjaXlbVjN5HmdDSf4DfklMe32fROPfHHeABT0gvKNWsNHClEaF5qVfP3c-hVLyRHmFeOT6UrplWYqJ0iBoAeZEyZV2JeNLrxV8dc5s/s1600/ex08-0530.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEidjO-7S83QZnBK5NgKDhPV5X4OyK8J6Z7xrUtta5WhqPn7AyTHcOBMU90w2C6kznaYgK7341f8iqc0HWQEXxpxBe22ES_65X4NYjjCesGXJrIdx5uoC_Xn9I_SaTPF_t-MhPb3nr5Liiw/s1600/ex08-0531.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh0G418XkY4omlHSYQue40c5_L9dlirn4IMYC1yJoJ4QGV_ob4wbV5hoVtoP4SyWYQu-AdtiQXbooeqyfOaycX8LC4TlJkBhgQ-9tusWBeRoLfwvBv9XjWm8ySy1If7Dp5F0ovxKTMTgSA/s1600/ex08-0532.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjsp5u_qWEGxNP5v_3Hgo0khVJmg9fEl0zxxqquLYDG6954lRt9mnw8hXhwKptpNb11XLGCjmyJVcCvsdCfekDr-I2PYm9g-YZ_EsFXFOy6ZJE7nvKSRQMKDnuOfJC_kJF4tOQyVSFstsA/s1600/ex08-0533.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiaycJabxhXcz9ZC9cUTl9BEdwH69g52NL5UzD2iRMtLQHerRQGEGuC1te0bOArauB4TgCLyi3dEm-43uVKpCDPm7_ESdnY5zynpA-kdSkZ_Ondy2snyaKFvxDvux_f1dhhx95uPavv_Bg/s1600/ex08-0534.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj8jMVuNMMrAQ-WVQHrjH6zIPG5ejxRjOb1OTBxBTBXXRC7lp1w2KYlsC-zrm-c4MZ5e8_awciGYiIuhsD-ZxMTlQbO8N87iLJ2Rcb4ro6pq7sp-K7zUR7-ggoL5CaS8omvTPhooKYBPgM/s1600/ex08-0535.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgGq1kb23jUYA50Rv6zmsb77Q0afnKdbd8dvGL85vM4dmofaq7rlq0uDvaRMoNEcjte1Oj7JgtAn8geXeqMI1DlTCZP9x8yX0U3DgJBK9o77XyTzweux6YoZH3S6IWFjPboVXxZYC_hzrM/s1600/ex08-0536.PNG" /></div>
Sometimes we will be asked to multiply radicals with different indices. This may seem impossible at first, but it can be done using the rules of exponents as follows.<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi1nWEzH1qBuLAIfswV5l1PuKRDB9wgyJ1wm8bjZLBcYX8VDdqex-X7RCZCoe0NDTZ5Uk7Q1VCwUTUzvrieyS8W6sTjBm1F18H3h54VKlNSRIZ27yDKBZeyyB_wxbNH8fTtXynlvFJg0j0/s1600/ex08-0537.PNG" /></div>
<b><span style="color: red;">Video Examples on YouTube:</span></b><br />
<a href="https://youtu.be/uPMspltvkyw" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqyce9j7I5LEq69QZxQ7vUytHb4KRBeAEjNhvho6uHTuggycV6yIL_SLaiD80IhcNr1_tuphO4IJhGCWHVgdXoJ5IYMiCXUOHoaXreOClXBu7mH2Y0JClVlw8sNYVWsKn1N-zumVW-HGY/s1600/vid-0827.png" /></a><br />
<a href="https://youtu.be/WjnbUSMu8nU" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjdavRMSk86QDuFV00Vvt-uV2g03q1I5HRgKvsYUW7uEc1xJnv3eH2DJa3JgnMBTR5MMDs_OaZpuVpgacFTYAoGUHCGntI8f1C3qpUkf3QLXClkFypuU3z43Qypheehn3GxoilhZ7YHw2w/s1600/vid-0828.png" /></a><br />
<a href="https://youtu.be/7iaDthlqI78" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiSJNF8IaheszwiOcJHeLxUjtFpj2su0me_8xA29n0QmpApK8KTBkeV42JlmOz-nkjVsnpED3Jq43y7SvOX48Bxk_A7O5NTogEqbi2ralz36eaOAB2rIbvfE29t0wh3n16nM8InPFTfLB8/s1600/vid-0829.png" /></a><br />
<a href="https://youtu.be/j4yU4jc4s_A" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiScEQz1dqy7_r-s6My_K7w06b93zDyXckV6iata6tUX2LDYzPLG2BwT9L6TylEa6WhhW05iMMDSblJhz5c8ax78CbeIdpkUGCu4mNr8PkN2MPPidNSUeQgVxiGlnP0D4uTRO2taCR7bz4/s1600/vid-0830.png" /></a><br />
<a href="https://youtu.be/_maq45p4NWE" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhYnWyYGZTzv5kNavaYT4t1JnL6PCZnz63Kjo9zwDz7z0jww4aF3qE7sZj0ko8Zd_IWBqNs4vN8yiHKAO-97GQZ3D3G_uw169meB1EcSQ5KglZfqO5u9AYGPtEmPuRD3d7kPYiw0Pl_k1U/s1600/vid-0831.png" /></a><br />
<a href="https://youtu.be/84ebXSjpcvo" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiFapMw7KhsFyKWASp9pvtbFV_CcfAyv8R0K8o6nAEbAV-ITpANeMIJrBWpPL_mYrZhyfnSce33LrIcPWtqUQBOXkL1l2epw5HbBXTNACnUxylLrK_IFuB7NIVPbsolIzl8WO5z7g06CJo/s1600/vid-0832.png" /></a><br />
<a href="https://youtu.be/K4aUwI4JMmo" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjfSXXP1QKFYa6PeLf638m7abjsWVetSmTMPhmWsYnsXee9n1hZBO3ZnXRyXal1BOS5Bq53MoTs4q9rUcGjwZSiRJp6vSBG0VoWL6nK2Gvrkw63O06lwSyKLWx-mIR8912VFYBHlZQPI3A/s1600/vid-0833.png" /></a><br />
<a href="https://youtu.be/kPAgqFrByFA" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUxo9b-_bhSWmWGxfoBFizPh_96pWhwP2OzGyNPwoakdd20bTX9_zUa1qsOVSjWeOhNwZjZyH2vhFQYjLKzAO2hL-pnNFgX_nXuYCOSLbgqTot8vGXRCIzdGdcjXE-1JDEukE9XJj-GpQ/s1600/vid-0834.png" /></a>Unknownnoreply@blogger.comtag:blogger.com,1999:blog-221233850938098964.post-68557629719784256382018-11-20T09:44:00.000-08:002019-07-27T19:43:59.692-07:00Chapter 8 Sample Test QuestionsClick here for a worksheet containing 20 sample test questions with answers.<br />
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[ <a href="https://docs.google.com/open?id=0ByXp6Ibu-fCyLUZQcU9kVGlPNkk" target="_blank">Sample Test Questions Chapter 8</a> ]</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiTj2ErLe6nVrRKNOJ2KEJXuGeXoA4FAMy4yl-zuq8EEAcm1Oq1pApfIdGhYi-FyoapCmtWzzTs5Y2k_CWjHLM6g9CTqz_4BdjbyXEJBqJgcV-mNpYGczcUzhs6NJXTUx2cN1Usqye7kys/s1600/ex09-0901.png"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiTj2ErLe6nVrRKNOJ2KEJXuGeXoA4FAMy4yl-zuq8EEAcm1Oq1pApfIdGhYi-FyoapCmtWzzTs5Y2k_CWjHLM6g9CTqz_4BdjbyXEJBqJgcV-mNpYGczcUzhs6NJXTUx2cN1Usqye7kys/s1600/ex09-0901.png" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgfFcuGdFI-chknoP9sfze0r3siH6K_b8sCsQRnpLvxecNgrJKPzQ-ViMS6MTv_xNpCaq-pLGC6zDfOv3z90a-m1T8735quBbe0KnUWdJPMB0aWKXFvNZfQ1D2V-30EM03FhJgvwjx5wz4/s1600/ex09-0902.png"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgfFcuGdFI-chknoP9sfze0r3siH6K_b8sCsQRnpLvxecNgrJKPzQ-ViMS6MTv_xNpCaq-pLGC6zDfOv3z90a-m1T8735quBbe0KnUWdJPMB0aWKXFvNZfQ1D2V-30EM03FhJgvwjx5wz4/s1600/ex09-0902.png" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEieFHGvjXmkb1OYIWorl1j7QrUs3M_EdGTEpH2v4YFwCCuksuv2mjRDITOKrKLudp1rtOXiybkzPynOt6OOjEnI2SsOthKYNlogs7F6nIzY03tS5TJMiZ9rWOfS7kD6F07BYwMDDkqbO9k/s1600/ex09-0903.PNG"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEieFHGvjXmkb1OYIWorl1j7QrUs3M_EdGTEpH2v4YFwCCuksuv2mjRDITOKrKLudp1rtOXiybkzPynOt6OOjEnI2SsOthKYNlogs7F6nIzY03tS5TJMiZ9rWOfS7kD6F07BYwMDDkqbO9k/s1600/ex09-0903.PNG" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_XxeZgE_d4L3h7USWtD6Gw05jKT2gKevouQi7BP1rQOihI_ZewbqDdqSVH9jgeQ9Dx0L2w6gWMAR6Obi3dYQJ6brgczqOjeCc5I2FUW8-GfRWlGYQwOOq1-wUgQ_Ltl6Jjwd1SXRccsg/s1600/ex09-0904.PNG"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_XxeZgE_d4L3h7USWtD6Gw05jKT2gKevouQi7BP1rQOihI_ZewbqDdqSVH9jgeQ9Dx0L2w6gWMAR6Obi3dYQJ6brgczqOjeCc5I2FUW8-GfRWlGYQwOOq1-wUgQ_Ltl6Jjwd1SXRccsg/s1600/ex09-0904.PNG" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEikcwO9zyFbXglNkJQTGnh7yDwHM_2eVN-l-sgj_QQZozsez7izk14RD2FHLjF9ESk4T48No6WLvMKBuFW6BC3t6M0MZa7CHpV2icLKqSutWySvzmcmMEOd3vWHEuLscs7HGXP_Agr28CY/s1600/ex09-0905.png"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEikcwO9zyFbXglNkJQTGnh7yDwHM_2eVN-l-sgj_QQZozsez7izk14RD2FHLjF9ESk4T48No6WLvMKBuFW6BC3t6M0MZa7CHpV2icLKqSutWySvzmcmMEOd3vWHEuLscs7HGXP_Agr28CY/s1600/ex09-0905.png" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCrQ5fEOCYtvEwwwnd3aWxssBHtyg7o6cBVSW6eNJEGb86MHC_CbBEAxO7Byqq1Quiz5br0kH6TP4f0WEll2fnpMHJdGm_66P4Ql7Tn1vFLDeLFpbQNB_J1rjDU8Ba5XbNpGz9anijIqg/s1600/ex09-0906.png"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCrQ5fEOCYtvEwwwnd3aWxssBHtyg7o6cBVSW6eNJEGb86MHC_CbBEAxO7Byqq1Quiz5br0kH6TP4f0WEll2fnpMHJdGm_66P4Ql7Tn1vFLDeLFpbQNB_J1rjDU8Ba5XbNpGz9anijIqg/s1600/ex09-0906.png" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_zjFuI5EKvnxW08kJi7N-CgnXfrWH5OagBi_eP6gZp41csLLZDEcexownnxEXHF8w2Td5-0zo4_8XyWNxH3PxNwaFSMN03y3eVUfn65lAiovKJdUzMZLTIGj10F1myf3bn66lPe0vR8s/s1600/ex09-0907.png"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_zjFuI5EKvnxW08kJi7N-CgnXfrWH5OagBi_eP6gZp41csLLZDEcexownnxEXHF8w2Td5-0zo4_8XyWNxH3PxNwaFSMN03y3eVUfn65lAiovKJdUzMZLTIGj10F1myf3bn66lPe0vR8s/s1600/ex09-0907.png" /></a><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgWsMIxbGV0Ps5qOtNL_b_BG4xJHgmf4ZTOtWn8QwX63_bSE_4lpNEP6dGtLywwtpPLJS5_yFsSsCwR8D-s3Vn3QFcuwLGG1agbFxDSPvmcmZ6wwqM1mcBH8QLB9tQBpUY_OnnVkjZLsMQ/s1600/ex09-0908.png"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgWsMIxbGV0Ps5qOtNL_b_BG4xJHgmf4ZTOtWn8QwX63_bSE_4lpNEP6dGtLywwtpPLJS5_yFsSsCwR8D-s3Vn3QFcuwLGG1agbFxDSPvmcmZ6wwqM1mcBH8QLB9tQBpUY_OnnVkjZLsMQ/s1600/ex09-0908.png" /></a></div>
<b>Word Problem</b>: What length must a pendulum be in order for one complete cycle to take 1.2 seconds?<br />
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(Use the above formula where <i>T</i> represents the period in seconds and <i>L</i> represents the pendulum length in feet.)</div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjGgHSt24qiw8SaMNSXjaTg7lyyZ_ETiaIGLxaItMn4UdUHq9Er3ATzrymXPXn-NDAgQo6-nP_OSUNTqH3OEg84CFR5NBUlv4UHin0AJ3fFLPVv7m9PBwNYxmPZQZmbTFrnK4plmRb5yIA/s1600/ex09-0909b.PNG"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjGgHSt24qiw8SaMNSXjaTg7lyyZ_ETiaIGLxaItMn4UdUHq9Er3ATzrymXPXn-NDAgQo6-nP_OSUNTqH3OEg84CFR5NBUlv4UHin0AJ3fFLPVv7m9PBwNYxmPZQZmbTFrnK4plmRb5yIA/s1600/ex09-0909b.PNG" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjqoZZHdUgHjkcioLOrr-elwnAjszmlYbsSyVBvHq1l81EW_FsMF1m9Nwr3TsRXvHCym-KxFKqxaZpHpqSNgfeVC3d3Abmf2hLD_n-w-e1BGeFT0G3gIcMM1TntJ6jaqoAFuC995BuJRzA/s1600/ex09-0909d.PNG"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjqoZZHdUgHjkcioLOrr-elwnAjszmlYbsSyVBvHq1l81EW_FsMF1m9Nwr3TsRXvHCym-KxFKqxaZpHpqSNgfeVC3d3Abmf2hLD_n-w-e1BGeFT0G3gIcMM1TntJ6jaqoAFuC995BuJRzA/s1600/ex09-0909d.PNG" /></a></div>
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<a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhWsOBExSK1weRfUeYdQ-lpmpKqqmd7b9GAusFdAyjkOfirs81xmEkbROuLT3XeXA214dPWpiSZ4peJ9sX3UhyPlus_C-_s2zSvyMTqbPwwWrYsqB-BFvbHm2zGzOkLlx8TFzzqCPbkpHk/s1600/ex09-0909e.PNG"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhWsOBExSK1weRfUeYdQ-lpmpKqqmd7b9GAusFdAyjkOfirs81xmEkbROuLT3XeXA214dPWpiSZ4peJ9sX3UhyPlus_C-_s2zSvyMTqbPwwWrYsqB-BFvbHm2zGzOkLlx8TFzzqCPbkpHk/s1600/ex09-0909e.PNG" /></a></div>
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property for logarithms to solve logarithmic equations.<br />
<br />
<div style="text-align: center;">
<a href="https://www.youtube.com/playlist?list=PLmN1jmOiJEf7RUvQsfl-FJphPzdfqywRY" target="_blank"><span style="font-size: large;">Playlist on Solving Log Equations</span></a></div>
<br />
If we are given an equation with a logarithm of the same base on both sides we may simply equate the arguments.<br />
<br />
<b>Step 1</b>: Use the rules of exponents to isolate a logarithmic expression (with the same base) on both sides of the equation.<br />
<b>Step 2</b>: Set the arguments equal to each other.<br />
<b>Step 3</b>: Solve the resulting equation.<br />
<b>Step 4</b>: Check your answers.<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEix_0lv9ZEYWRO27sIjv_FKqP9QNdfFUMCQMghc1ybPx-AI0f2sN3H3mo_4KdvgbkpOW7ArGxCIsXvQRHSQNEu88lozIXYIfhQiCZchzQ_jqVaQ55F1GVH57-F27g2EqUkavTljKS_uDMg/s1600/ex01.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgNz5W_swLMP9_o-mJSJTOBrM3o6cFK_kJq-ivQgKfkUwQsj4Xj7Cm7RyBILZGbwHAk4GL9VDfHXm4GJxYe7seV-ExOQPRgOXUSI6yEkA9Pzdo6osa76t1aAMjcU2DQ4eVOcqW5Po_cEIA/s1600/ex01b.PNG" /> </div>
Be sure to check to see if the solutions that you obtain solve the original logarithmic equation. In this study guide we will put a check mark next to the solution after we determine that it really does solve the equation. This process sometimes results in extraneous solutions so we must check our answers.<br />
<b>Solve</b>.<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvvPRsh-tvafAmAUuoTeXzh4WwU5Ewl312rfTcZvLyOZH_rFntJFftF2wtcx09pGIcgzLihLIGN4xgjTBVdnqs7QflbX36U2JH14s8WaCvrK4UtNEmHsmxCWxHoZs2IGI_UW7lVcN_LXw/s320/ex02.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjEcfSDF4zQTEu9Yqq6B9UagjZGByubKrmF9v5x33vSXXPePQgdT4erQgVo4Lp9TlrgoCBe9CeuFhDZVO0lVANFgadhJBeXhNVbJXfqBGYTu3gLsAQses8nVTVHwmtUCOoFsDnhZGl9zqc/s320/ex03.PNG" /><br />
<iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/Wcf4SajxQl4" width="560"></iframe></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiPQwJbY9btyye47S0PVtJBSVBtOmW7EtT_QF5C9F7nNWu41u5W33eJ8m9z5XUAlWYyB-YR3u3A2Uouzcasgar6szmtfSQAIEZ774AuniKjRvQtQ8_3Po4KgIjuW9TtcYRZhWp_Ue7r584/s1600/ex04.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgTtVNnW7P5cscNSnRcx0M7wiqpJ9hedyg0Fnlr-EcsyTTKfmjP7MHINw4_TzFdvtCWiJaaRStwOhbrvM00yQq64Pu0vLWRhXNacSfU3h6FK-9iOlvxLpAaENSI0qx72D7erb9jlqhHwdw/s1600/ex05.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjKuv2ER6SChyphenhyphenKGywPH1zJkP6vf3_eiYQLP2Zx_ev7pPsmqlNBvS0tZkROmiIRfqKm0ycgrOSGcsTe_7qT32_WIa93MHL1bMZdLx3YVFuXaefMHGaChwlx-fO60xH1UiNVyYkbeAWpYtgU/s1600/ex06.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh3QmEguhV02u-Om0Fm4Zb6qSktDpwxDUK16_NwvpPe1VMWOKeqoDgjDeq87TZyMWWLMLRlf2kwU8SPjsRqF3uCaObYawVXyPWFB1fTc54EePcvWXRBbMeQbiPPoyagpCLX4PsxrDzaweE/s1600/ex07.PNG" /></div>
<br />
Of course, equations like these are very special. Most of the problems that we will encounter will not have a logarithm on both sides. The steps for solving them follow.<br />
<br />
<b>Step 1</b>: Use the properties of the logarithm to isolate the log on one side.<br />
<b>Step 2</b>: Apply the definition of the logarithm and rewrite it as an exponential equation.<br />
<b>Step 3</b>: Solve the resulting equation.<br />
<b>Step 4</b>: Check your answers.<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEieDBxI_A71-J1x8tVurAU48auijW1glb_ltpSQKQY5prSngsaTps99vJ-2mzyfYzE3rtRWOHAy3I6cHhlPDjdjf4CbVUcEKD8ek5cc2mpOQ_MOhtIOWUqXXZRnLuMm9CD0OvlVNKrRQTk/s1600/ex08a.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgnvAi48QF8R85eZfwNwcQI21ZReW1JTsiKxnNB78rrLFhXwycHaYudcaxhMRrUgDxzuDjwqX8aWEBItaamjB4pv6qZIKXrPYbwox_flkUwW20gchbf8rc3EgMmbzRcPIUgLO13IeMj-ag/s1600/ex08b.PNG" /></div>
If the answer to the logarithmic equation makes the argument negative then it is extraneous. This does not preclude negative answers. We must be sure to check all of our solutions.<br />
<br />
<div style="text-align: center;">
<b>Instructional Video</b>: Solving Logarithmic Equations<br />
<iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/YWuSBWyWh6w" width="560"></iframe></div>
<br />
<b>Solve</b>.<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEilrXobP4zo-f3oeJeF6ePksHCySgt4cY-sKxCb_zSBSvlaCPFs1mg70qQqkNkUIeaOcHGOz42QuROXUyd6VJhGEgJpRVMYrJYtXGH7WVddkU8cRTKJ224fdSL2BXwjbr-WvQMRgSNyWm4/s1600/ex09.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKQu1WgrWNFLFfvbAa0jTT-X0TFsa2FrnWWLTmEjPC34hLelt8N2iANF_7xaFIPzzKPYjbmkzNVmE_SeBgCYM-6zjPfBTXpit-uSm8ssa328JTxuAlyJfARTPyCpyxLrPKMcXoPoJ0B5I/s1600/ex10.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh-LWlYggxDbvGAhkOWiOfuIWzj6waLfNidbnOMX85QsfQF0zJmf4XvJQEnNQA1JSB_FohLdU36BDk-DokWf1ZYuH-MZJ_ZK0rhL7S8WwyZus9Bm6V2uHVodprBUv9FkJIponCCzZeeaPg/s1600/ex11.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjr4oSyYKem88_oX6A03kt3L41LZ9uFUTpHywka9iALtDOHNoqtJ8v7YNgIu10XNkhqFRYB3YS9pM1wcWhYaaYAl6mYoZ2qhkzJOkm5ty7YpTiGQs7nlK3yDJQ6S3y3u2FR8n9q4MVBRSQ/s1600/ex12.PNG" /></div>
<b>Tip</b>: Not all negative solutions are extraneous! Look at the previous set of problems and see that some have negative answers. The check mark indicates that we actually plugged the answers in to see that they do indeed solve the original. Please do not skip this step, extraneous solutions occur often.<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgKaB1sNPk-7E3ksuVzdKkVnlLD0G2dyAzUOUBS7YaLix8xu2migDlA1a6wGDx1rpU-CJEX5F5zkwH1boVcNqQMXUo3oW4lEfKareEl0YI0xkltqiydg0tH8oMnd5lItEgSq_N8DlMtGfc/s1600/ex13.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiKVzUCz0jxEawHyH4AiZj9eHNKVTf80X0nHnHPvp3Q0TPNooAs_s03L1ej6lgPa6wLomjI8TtmEKO-fN55rX87qISL-HkYO8Z9ZkotaO5M76Ld-3G3RS-HtMzsUQ_wmaQqJqqhTaPS0fY/s1600/ex14.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiXgjsiXOvQSW1uSAGbIP0SFbGAQycRzrNBz2SIc7UyQADBbKVcsUbgXgNLBZ4IRph28liluuIoq_hBd_Fskr2MxdvtPbcbEKyolgxERlDkoS8_o7LSoouJIJzBO6FtRfJZyaLFo8cDNQg/s1600/ex15.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXTG3yUUkgCJXLycor1nUADCvT4bvSQVbBbWpEB6be26OGMhTiuNZkgqDRTZwh_FQphM9EqNXlPDe88CdRpChqi3PLO0vkc7ommnRciZxoAitXhd6tLWFZS9_o3N-jcEuxopwbvEJPndA/s320/ex16.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYBMQWIyWOw7W7QLrkvN-hyzrirZjEwy5R8McualRACwLER_o6PqdyuYMshiR4bMpyGj11OqxUSltSY7lexjlRvi0E39UNoRlbd-rwJEygbMvvv2VB-04VVTowynrMNL8kXXE_ySw0I0M/s1600/ex17.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgm3SCBpUquHQP5f6JHHTiurezS_QEa9M0nS24rQK9qApUh0O9aBrMyQvufkmb7SGRwmzhICQJWnLuWE2FupWYfuoxOSiTTXo7UBMYbBkJB2Puh-bcP4olmUQYou9pPhDwM1RsM9OxIJuQ/s1600/ex18.PNG" /></div>
<b><br /></b>
<span style="color: red;"><b>YouTube Videos</b>:</span><br />
<a href="https://youtu.be/rnjhGF-BWho" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgItGH0CY_VShqlz7htTsjRrm8yhrRVhRs9wIBE3LToNrtwLrWQFP53dF2BwB9eztS6TkDX8IybMkUD8TlotLhyy0MKlJQi2jyXwCWuPIH451adE58_Iaxlh82HiY3glSxSJMnZcE8w4K4/s1600/vid01.png" /></a><br />
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<a href="https://youtu.be/YHkZ0DekF5A" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMRkJTz-LXIO7yNMiQSYRX9id1qoFepvrfNxiPyouSaTYQGw414BPzE3fMXGN-HzusRSPToH9dUGKq945bPihh29sRUs_Straif_FgtZDxNnP7zWh0L_DWECDjMt87JqxoAHzjs2MUEhU/s1600/vid02.png" /></a><br />
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<a href="https://youtu.be/Wcf4SajxQl4" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhs5Jg8ZHkZ_vUtT14GItI_CNLW1q7KtvYQUycjG020oFMIy2d6Igro_c389yvtE3nRJydim-riSeYqN3jA_jTtDcN2qt9yx2v6BJfJcKGslFFICdzB4YpjyK93IsA3jAu4sVD1I49P_mI/s1600/vid03.png" /></a><br />
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<a href="https://youtu.be/gKJEThYc9dQ" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjMqHrXBFl1gA-ogPpt3pKfPcr75nYP9kGHnOwHwseTjIV6ZPF46sQPMJQSU27Esjsnj-OYciK8nMjXMLDZ5DyDfHxOLNPxzmRCZSXwMwHQf0uDfMyvkGaVkAzsNE54HIhpPnyw_70Q_sc/s1600/vid04.png" /></a><br />
<br />
<a href="https://youtu.be/F864AtAYRL0" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEihqEKz-PrLvgt4T0HSUjQVmXgnFzKqetFugVa7shvt2wAUBUlt_VwatcjPx5zCRn-9ZnNDDG1j8vkr0qLYsKnaQW5ApamHERR5ZUkT5uGTcBqaaJmRJFgHaiHnM9viCk6B1BmasZUeuQE/s1600/vid05.png" /></a><br />
<br />
<a href="https://youtu.be/mLPxqGC7oL8" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgWsBo2qdv2HARDtAHBMBQi7fZAd8X6oO_Cof5NeVr80Q-3OY_pmscakhJj_0kjuA4kitEP6Ws1_oNK-M_9UJxruA8cd4YoPHdjXqmt93q8OMDzthWUPqXS44LobnSTnzEGrsRJ0NBQRcU/s1600/vid06.png" /></a><br />
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<a href="https://youtu.be/Oz20edlRDow" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNDj-LbGY8E2NHA6U5AKaln8yfxWPUVP01EZBluKcLKFQXythbnufXiIW4GAMoR0YK-yOfQaV0RYyQMtaG-6MOuIOqkD82LQ_d7VnTjx6mD9U8a53jH5sYP2lgqv58i1dPsqCKwTRjr40/s1600/vid07.png" /></a><br />
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<a href="https://youtu.be/YWuSBWyWh6w" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgw3IhElfbZIVJeirnCLhEd0D0paWYp72wVL2Ne67kJt_IM3JuCPGyatxcsYpcP-I-PeooW5cTjo-FGuawt-ORTSFf3LaRyB6aj11yRvOuVyIaTSE2l-w02Z0jWhTuWl7hGhJe-VbxB4IA/s1600/vid08.png" /></a><br />
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<a href="https://youtu.be/J0uNxN2A7Y8" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjpXJ4QTbh8PbwB0xVfx28JTP0RHpLutYDI8v3VZdkfAjJQdiKio_W_kOkyTon6UdSfsRlikNOaa80aEZGejplLmCQ4PHp6ys-Z5n7j3xo-XcIZhn8oOiIYrTDwNoD1REcgPpFj2TATDzY/s1600/vid09.png" /></a><br />
<br />
<a href="https://youtu.be/9e5XWUn4dA0" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhd78ArcXv3Q9vAGHemDtUnM1OLpZolJKwX81JxOmJT-ZYkfsGoHh8QH429hC42ZgVV1YtuA5zbCJBiFKiRk576cw7Xh-PFsryaziDO0Fv3hLj6VcfGk_8YrMROPWXcUCOLYoR7otREDy0/s1600/vid10.png" /></a>
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Reddenhttp://www.blogger.com/profile/02231088988057526726noreply@blogger.comtag:blogger.com,1999:blog-221233850938098964.post-12304537598598127162018-11-19T14:14:00.000-08:002019-09-15T20:26:31.169-07:00Multiplying and Dividing Radical Expressions<div style="text-align: center;">
<div style="text-align: left;">
As long as the indices are the same, we can multiply the radicands together using the following property.</div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: center;">
<a href="https://www.youtube.com/playlist?list=PLmN1jmOiJEf4hOOt9DsjxGV99vBDSMsgk" target="_blank"><span style="font-size: large;">Radical Expression Playlist on YouTube</span></a></div>
<div style="text-align: left;">
<br /></div>
</div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMpcXM8lH3Ey21KPphd9-J4r7vYPXW2tw0ZTvsoyh6UnVGsY45n5Sq3MssmP3Sb0HfjZGbL0_nW0jdA1kofTNe5wq_YLl3gGnXT-WhIADGssuy_deBz6YhyphenhyphenbT0SXVrcQUsz0ku4I6uJys/s1600/ex08-0401.PNG" /></div>
Since multiplication is commutative, you can multiply the coefficients and the radicands together and then simplify.<br />
<br />
<b>Multiply.</b><br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg2hSjNGpwy1jUAj_kfMK91Ck7ETJWn3wT4rue0-hawLnPKKnx3stSK5g7eueJcvlE6zYP9Y4DzSDnRkONGb3GQrY_grikXLDSlO72MXdkeDpnZncuxNVdMa5_cwbY0r2TaoHpQYV0QLko/s1600/ex08-0402.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgZiK32YLPuBPxXNfKcQeL1gMQZ9j7oJhNWg5wq1Q-XaaZfoqV_at5yvlozDcnjBcOdc9x2Sgrt41W0k-w1hhGj5IRgVdvk_MXney5BTwaxwsmGQLO8BfLZY5numBsup25NsXaj2i-HwBE/s1600/ex08-0403.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgIx-eX2DxJV_UynsPBjGMhKOW8bzbxdDVsFvrLSLSR1JJoYXB_pdgL4MwWkWJQX_n87ET4UZ_cuZNlowf9Wp3vQmCYQgQb7nvlS4wJh3uwFOnyKeO5zMv8OhN3LHLwCNBGC75eoaNiMos/s1600/ex08-0404.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhDC7obX6Bdav1AURISW75EcZw1_-pf2TM6cZMRRwIU5yfPRs9wJQbj0M9O8F3hIOlmqIoiHaSnTUnVhxFXMGfpXlwlIPnvNkfpz_C8P32P-EVDX50y5mmGHS_GD_TxiF5Ikhig5UpyWFc/s1600/ex08-0405.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiVhUM8FfdQBnOTVGzGeZG-B9lWHaYW0MeUyc1gA_adyNIGvtuauZm5QSJNDrGe4rkllDjWEdq0zVGmjWdUrEYCY4d_CeR-jldt9iI2Hw6LaWZTSYWE0zzgUUI2tPgiOv7-jwtUzbDH61U/s1600/ex08-0406.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiFz7fPv7B9xditECAOS30ntl0VBMyKGWLLmjSxgE3HRdMSp17sUvWfrUnbunzJshbcyZyhYBckbGBnQ4VJIFArzrXil6p4AZoJ6I-Q97WH8on4RTkix50L9O442g0JlQxE10D2bdrvBFU/s1600/ex08-0407.PNG" /></div>
<br />
Take care to be sure that the indices are the same before multiplying. We will assume that all variables are positive.<br />
<br />
<b>Simplify.</b><br />
<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjP5G-XsTbDzy3OG9T6exeAYGD_COWlx930PjV1Xi0S1D2p22LZ53XYknJ8f2kOfrtlmBVgPxbhlnoj6Hwdr1vDHUJufSpgqjG_NYqvM5T0iwzwjoiBLmyi0aXz9GaRVbtZWGNfbWpZBGY/s1600/ex08-0408.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi9zK5JeHGh9SX6aDmcMCtDo6zrvLSDkrmp2K2q5NouIa9v77IW1nrUfTLxsQoK8BO4kOgM5LgLD5eqTdi5xOvF_nAUAvt6yMGMt4oh936epLJoRN7rfQxa7w-h3UeohCYwqpn373xl7WI/s1600/ex08-0409.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh95mI2aL2ATY0j1q_hJcmqtfBW8q-pSEegWP1u7POJdxp-oZPSFSROUabkSzCBvUFV3CRs4K36kjxcMdZJ90pYG3P_aTjtdIUDOK-pwgjc_jMN3x5oTxtkPVmP5r2RYv5hLijIniXh590/s1600/ex08-0410.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEibHnIwWWhcoj_-_H4uUmvjEzwHkpq7f8FTawbpPB0nwVMZCVe52MbQQdOfh4_PknUiMOIH67MCC16sTwJv4JFvutswV88vwviSLnpHk3n1QtMx3PgTqy3v3xdODLgNKaoY5DoL8MAZXV4/s1600/ex08-0411.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhl0FpulhQdZmCA3s3uL-B1dwrOYL9QQEEy-4gozjXoM4N1727rv3avvG1Akn8QNDQxcLiMWJd1a5TuGtjCzAI38dKjKsmnhGZPJ3UIXyVJT0vW0-_m3A1X05NiKUHV0j64XJAdpng8Kuw/s1600/ex08-0412.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiSp7KPx4Ge0XVUqV4xCzJIHH99B0YhSFDBhuKnI48risgwtxj0nVe20vFPEPxPso7Or-oIqNCBKLWDPru0wIS5PmuxNEtC9h7b3A9kkXSzDxppscvzdw2W-fkKZCvrEx4-3kZKpQ3rWG4/s1600/ex08-0413.PNG" /></div>
<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjOCAs1j96_K2YtukzITUfq6AKaJfrCAWIsF8eqXiq5z28VGygYdV5xMEHSyg8-ivn-v5YN5Gq5nfStextLY7iggcI1FwdJo-Z0kqptgjrtV1rRac4EkBCy19VsrZ_hHpqfAWOkChD50Z8/s1600/ex08-0414.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhXkCiGcE6vcKLV1bEZPSrjZeQZXWUmtuBx0fpOAoC02oppgjBB0Eed9TWtruTXGmXxt8QUSx1fdudhiE6K5yVTsPQ0HDp9G7qhkmAvcpHtCsZIVSvGjObQRX81uGvJuorwwcCdwVyp3PU/s1600/ex08-0415.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh1OPLNCEJi6ve6Qzky2_fL-yR3xwb_DzOFYUr9xLlmYYaL1fGWPiIbZwBJq_Bk0ByrnY2VHVbcWiK4SQDA3ViL0BaEcUhWuwr9RDOcBccdHMUhvyN4-2fHhaRr7eVA99Ugqp28T9riDUk/s1600/ex08-0416.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQeZgz0RRGS-2Bw057TKVJ0mhOOrmSIA6Uu0DUT0ar3LrcwaRzhPZ0UabzlMV1LPR5md2NOpFSGUsxi5IKqL1hXc5ZglXMyPJLEmNx_46WodjmYTMz1BFtx6dlJq40sgQ_WK_W5zHuTWM/s1600/ex08-0417.PNG" /></div>
<br />
Divide radicals using the following property.<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjh_iMytnDnvKOmXQgplv21U4L7eftEMF48u4kGfiVUjYoCMxIjLxxoyvsj8-Q8UWE9ZdJxm92EjecAqjxHGFsADjcj7P-XMhYHruuoXOukQ5xDPRr2OKBXJ32Y4zC5xyXOWJXeO3rb5e8/s1600/ex08-0418.PNG" /></div>
<b>Divide.<i> (Assume all variables are positive.)</i></b><br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgKvHNHozFoeVqvZxCOwUsxLiCfZvGCXLLj1MLGY7NPv1JurUGaatkADH7E7Nbk2ORlq43_rsa2WbdgyM4D7PW8nkes70czfEnHERbUgEbM-JC4gLMI8IScHlo1Aalm5vBCNn9LHSwkn9c/s1600/ex08-0419.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEigIGbiUEiuljHkoow_msE3zNRq-spSFsT0l7wnal460T9ZqdUBbS27PbA3bS04AZoboNYyaSyYKsTo5KH7WAKtGWei5OFbtqozIzxnNxFNMG4iXc_2kROp6o-7sr10SL-VWHb25uzH2wE/s1600/ex08-0420.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiJk9GLHDfzCAEVo7vEekSgKTw9LOhvkJoGju6L0WwpFDSgXYrnpu1mOf_CXDUNnYSum4YXa1JZHuoreh4Hh1PrFhGXihCsrIxEVN8P8-u8MvBVfpm2_V0guvwFkXgPgJK0Fjh-w4yGpmw/s1600/ex08-0421.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgBkCykMr65re3lxD1CrVnMonlZFvo4ISo1YJF4BAnfDTskOWn-J9B5psOqtXSyOfIqyKZ9N6_xSxVlhfIzz9oc-DqEMU_1cjqDH6I_NeTgN5gMBJUlZSesR24vACwUNfRU7Gbo1q_TgjA/s1600/ex08-0422.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgDuUNJB-yXpyx6VkykzYFe3BwVx0fYTP9-VRhc-XH_gc2WHznOccUNhRx_bQJqAbKQy9acMwUtaSW8cPDeTjrItklaqoG9WX8PigR4m684ApktQ4iKN0VUpIg1C_hyphenhyphene58tEOZ4YP6oeug/s1600/ex08-0423.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiwma7rJb_lQXmxHAfbWyi_YK06d1RTPLptDJMIr20oGj4W0QWmocw2_rSSycf6hXbLZBKcyH8NODxgIK6y4xxhHb7TMEq9iMHspxexez4FpcjRd2mYUZ6gFhr1EJ5oQKzshBhEpkjv840/s1600/ex08-0424.PNG" /></div>
<div style="text-align: left;">
<b>Rationalizing the Denominator</b></div>
A simplified radical expression cannot have a radical in the denominator. When the denominator has a radical in it, we must multiply the entire expression by some form of 1 to eliminate it. The basic steps follow.<br />
<br />
<b>Rationalize the denominator: </b><br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjkaWKKyP2HIDSyxDLq2P-Dyd59s-rs-C7_mSsadOkDn7Z9R4eW-E1pac15od7PJyXvvynP9hxOQ_LYW02BT1EyiSwbbKyYuVs8LuQTJJVCnGWtCI5eOXSn4rW0pSQDHXVP6ZXs8g8rde0/s1600/ex08-0425a.PNG" /></div>
Multiply numerator and denominator by the 5th root of of factors that will result in 5th powers of each factor in the radicand of the denominator.<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhiXfN2aELDudnAYQBd65y2NG9zPE0GFar5hKxKNbkSLZpKu84P5ZZQw83NU2dExL9vr9P4xfL72QO21bNRVJCw_6VdsVGXEWIgXsUJ9v0qS0psJyCl_eunELxgccQfuQpTRpttp5DS1Gc/s1600/ex08-0425b.PNG" /></div>
Notice that all the factors in the radicand of the denominator have powers that match the index. This was the desired result, now simplify.<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgOtInbFgp95KqvxZr_Xfg8tBaHFOcslrpdcA0Sp7zYT0BAokhPPoEPI3jPiYAVFvhRBhG-JCy7vFqpRzzT7OoF0ZWK1BGCSPxhXztEoEkyvXBqX1xQ80by8XwNzxg5bfv3YbEAf3ZRaD4/s1600/ex08-0425c.PNG" /></div>
<b>Rationalize the denominator.</b><br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiBnZQXNArUjlHYJw0AOSzhlshWbS9QXvHuIFx9AMc0dyTwhDhtNU3tIR9HvCXNQSxX8pvxBTMeJSzVuqz5hsqdJgMJsO1tIVtxUIkUaRfQ1SftOfIXgyAepW2m3wMknnSVWmox1GTOrOo/s1600/ex08-0426.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh0b7IdUw_hiGMVWSD6qIbByDzHoDd8XEmTC5A0K2wiKDoe2nuzQCW1XsO7LVvHD510wQCBi4Bmp7u-fnMOFVZV5Fw8ZbOjBFKpC5s1KYHLXN83XXmqJywAUUzhXMtpHQbLIzs6JHNYwSU/s1600/ex08-0427.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhy9iD-VDzuUUuqYhilA2ZYwPeH7Ivj9JrKynNfCNpwF52apFpIJJKJflYg_7A2_lrvn2oK6R4hyphenhyphenLwT7KRjQmyNozeAWnTcQ45k_1zXqzdv1rITi3PRWNIpaHDqgvh94eGweOSvqDzXNjk/s1600/ex08-0428.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEingSNchruAIZQQ4jIHYgrpJNKhqY1Ah-Ye_DFSqJBRt7niVVuISW7nMBtEawgqSrs3j3aZMqbAp_Lnb5GdlulLk2kUQK9j_Doapiqrhb2DcdydJtQJNc15dAP7Rd3DciWltxqCVqA_Lro/s1600/ex08-0429.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjEK_jV0S4e0aDnzn6wV8FWRbYPxT7mLeJpeErzXQvaU915X57eUP4eMAAIam9Mnh08xP1_EBTi4U5i4GIp9cRM0DLwx1l0IPcrooSeWDezTmqdqdnUDfJ3-uiFVx78SFjh9-N7F5yN7E4/s1600/ex08-0430.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgOsXrPOkwOvOYVe1jBaUkAQsY-XqYMFHg3OFSo2kg9ArRd2eZtwDNGAwDl6ELOiDt-ySk8xLFLsYTVJj1MvM9kckBPL1diZDalnh-JmOrMntyp-T8m6r6LyIBO5IBD1AXbK2U59ycZWLA/s1600/ex08-0431.PNG" /></div>
<div style="text-align: left;">
This technique does not work when dividing by a binomial containing a radical. A new technique is introduced to deal with this situation.</div>
<div style="text-align: left;">
<br /></div>
<b>Rationalize the denominator: </b><br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhru9dAcYlrIVW5Fu48H_wVcVdXxKZ0w-qT7XCjFFIZL08bLF5S2YO-aXFVksTgvqcrkU50upQL0ZYxrjv2fSjOb4rEZ43EcPeFfV4B8fzs_oiwjMChagA-QciQn20z0TyORsGhUu2luc8/s1600/ex08-0432a.PNG" /></div>
Multiply numerator and denominator by the conjugate of the denominator.<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh4TsAA-JPwJ6tLXZYsQgBwybITcFqGLyT55YnRKK52RxXXiZkfvBQXEv8dlEILwTtFxt1pqYGNqV1iZplx5pJaRF6Ye03sIIcQSWVmqyuGPK-Jr9R8pS22rBR2XEZVKmGZRFcoBZumQ-I/s1600/ex08-0432b.PNG" /></div>
And then simplify. The goal is to eliminate all radicals from the denominator.<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg-uV5GYW0kLs4Sk7im04ra6OoaXlgGFDWamtQcm3_JF7-XeBmjmjwWfbH38cGfF4EE88yq85Gq8gi069m19JXETfDWG5frjbMZzPPycd10lRgMf7n7tbEqdC_26F0v5fuB-uaH89Lxph4/s1600/ex08-0432c.PNG" /><br />
<br /></div>
<b>Rationalize the denominator.</b><br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi3iC45dAkvjq98lkuQ5XP1yJJAtW8OjzJQ7QzabreGGC0mFw2lTRrmPEZElLFO6GAKdnnHDXGTSlpYR_aFnrrUnkqyGwxu94RZKz8PpYGgjvxwH2PFSWQClxjBvvXtbtJqufJPYCcY27w/s1600/ex08-0433.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgNxQcCwLGXCCJVLDYHqjI7WDh9p8onoj0tXEFFIWnPBvMVXKJd5zCnspQtJO_CBTyQSUHft1_edPj25XEgTYV7fiv_KQPGgsi9X_jkasUp2_JyEt8jpZmFqpclyTrec0O5jLoe58eQGKs/s1600/ex08-0434.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWMuYRlhjuGQsZuXfAE15w3hI4FWuHpBW2qSdeks6DFHQDcFs1NQJj5qVThdjsSLc4dHzC1AFKz3RqycFxkUajUZz0y06NsDztQ3-eKffRIzFW276Xml2CI6B5SiZ3g5DJrVzi1gQ5og8/s1600/ex08-0435.PNG" /></div>
<b><span style="color: red;">Video Examples on YouTube:</span></b><br />
<a href="http://youtu.be/VbqlsE8iSpc" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjjiBZP8jtTzbRLIZQMHyXVBhts42clsq5ARdf-F93Bn_5YElK8ikb9XKwWasgczg5LQXisUc0iRZSXJKR9dRMtNYSbC9haJVmdtXiYVQYEo_bPU5o9QrPKB1K7gZyGVtH2HbCtNhJDhm8/s1600/vid-0817.png" /></a><br />
<a href="https://youtu.be/IiLYkcUBCi0" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiV8YSfVFgJcKhZiymyFjx6ke5w-qGhOzEnq0yyiBGMxMw7cA9uLhGSu4DMCGH74iipG3eJVHD-3oyaF5sUypZ7LC4HuG3xNxNhKe00h8Ls1vLzAp6lSTkjP5kROACR8-dg-t6FUmp6caQ/s1600/vid-0818.png" /></a><br />
<a href="https://youtu.be/8mXLhc9TqQY" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrbG0rXuJHmHA1lhcyVRBYWlxZm5MmPTQ7BgAaG7EPGypMuzZnMjnhI1xwriKtYxHuCl_lSfdc5NC5hRwKLiKbHN5mqAbf7SYTBRb1Ml-nxGbAYis_eEhmBBHi3heXywdmjsRZiJm8Alo/s1600/vid-0819.png" /></a><br />
<a href="https://youtu.be/qht8nT6g6XM" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjZ6JS4Eb-WgHjtF3w0XYz3tjn_RAQOWEDZHkbK1D6cetQsnLzmCRbVIqYDXwDd8ddLiwvREBWfoiX4LpCFKYNeNj94LDnNDAv18wK09o9hpuWMysdpu8Zc2J4z0oBftdgn5aMc9hCXcmg/s1600/vid-0820.png" /></a><br />
<a href="https://youtu.be/OiS1J2fR9Xc" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh2Vigrs2TWfsSCn7qoJdI6-E0-Jl9hJhBaGBGKGVHJS-p702tTB9KwIiXDjiw_b58GOfG1z9csWq4hNcKeqFtsQOK-tjJMTGq4Aeaa35UzofvWvp-tPrcLq71ZSqL4eOIDKS0sg8kxs5E/s1600/vid-0821.png" /></a><br />
<a href="https://youtu.be/mS3pDb_JerE" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEik6kxNLjoLPSR0sJkw5uYeQz2JfUG7PRzSYByP6wQemPFh3LWs8VXpmFBRKNHZO_xeJcfPcHmV-m_IiOjoD-vFysdY5F95eKGZ297zbuLcTnll6H_J_eLbu4ltTLVf9QJZ5SxEBDl6ijc/s1600/vid-0822.png" /></a><br />
<a href="https://youtu.be/B8i-9rZ4l0Q" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg_1DeNoCngYK-StPtSTvdNh78mRE0nl7-p4HBg4dP-UHadbS28J2-wRI_YseO4z4wD4CsbyLxuDWswxsbvrUprPov4q3V-fPgR_j7dc3gm80dJhD2pDVveqpjNvCnVuQu5Zoj3WZ3q_0M/s1600/vid-0823.png" /></a><br />
<a href="https://youtu.be/nmKmWbXPwjo" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi2NeZTxvoiixGC8eWndOgHlXBLm32aAn8nTHOKOL0wTx5FLUO8foD2S5VAPtIC3vC2-923LiZr5VHcqVfMN-Xo9Ysikyj6821HOhto_heQswOTo9QlNozKoB0nSK_jMHEZmqf2ez1GLAg/s1600/vid-0824.png" /></a><br />
<a href="https://youtu.be/TPUd_gaJDNU" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhLoNs4GIQdccocex-xVaqD6iDqAuF33ez0oQWj86pa-i6120qfQnYzcOV653dvdhX9iJ9wCrv-jWK5M_hyESK7XxjbuOiBRCWOek8HPaIHb3ohytNMXzO9UPManu9EPCLXNizctrHZHXY/s1600/vid-0825.png" /></a><br />
<a href="https://youtu.be/scdd3Xfdpeg" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgHKcU3mNDUUxwGc801JjHpVnTlJAzBHkjhyphenhyphen3UjT051udyZ5Ohjyhdtow0dmsp_3fhIQITV2beOZ7b7vIVc-ZXYiQv4F_TKG-e7zHF8WFtDNkSw_KTNzj87W9V-y_4Y8wBdNZIMR0zMuiU/s1600/vid-0826.png" /></a>Unknownnoreply@blogger.comtag:blogger.com,1999:blog-221233850938098964.post-21890358762087237902018-11-19T13:28:00.000-08:002019-09-12T10:53:04.485-07:00Adding and Subtracting Radical ExpressionsTo add or subtract radical expressions, simplify first and then add like terms if there are any. The radical parts of the terms must be exactly the same before we can add them. (<b>Assume all variables are positive.</b>)<br />
<br />
<b>Perform the operations.</b><br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgVasYAlB7oAm8qk-h7ec5hAB_irA6dwZWpB31pJ1stfoW91vTnqxq_5WTxxmZZQgcetOzsbkHc453cokgcES3VzSfEJe_9jC6cKcD5BdvrYUWgCBN0Xe72IzBrdFvI2V-yIH552EIXZWY/s1600/ex08-0218.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhCtO22bWSm8kkkjdnzp_qxKqSXZDhmQXmVBobJlvyesJ34orK4PU3d9z-bz3jDXz00AshbADfnfOMtHRiXOQ4AQ5Jmm5x36TJ9JQYacn71MZxV5OEsdpjN-sNth9qmJG6VjPsJ1NWk_XI/s1600/ex08-0219.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiqzF3-6cFCVOQx3mhHeJlc5sDKm5x06N1xc55OLsrqKNgIysejvJBYKzlU4qq7VVJfD9LHvTzyGgMg-V_bEcHrrPm3IqHf2AZD_BDZIr_5eoZlXhqFqjdRg0ePLm4udKf3-1n_59c8M-Y/s1600/ex08-0220.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjDKGaLvJ-1HWIqD4Qv0M9V8G0-u0nLY_r7ZYQHNxp68Kfw9-QcZ0witADj-eaoKmeSRztLbpBURdQfTxcCv65y8NR92YLIKVISz2Fo0Qdc3tE8FdnDny_1EOWUpStQ_x0pADUZl_hmYnU/s1600/ex08-0221.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgcBr5N9GX6MwgZn8Z5xWIJUm0iov19RKvNs5ysAPOsNTghUABh9kJQ78kxT2_PUnVK-UpmqtqNVVwNdOqY7i00yFWtvLa3SD4tPiPvNBM1ljiGJr-BiJ-pCg3chnnirM65IED-p8Soa1o/s1600/ex08-0222.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgmEgx6K-VhdEsZCFXo1Dbv4x_SjddnMZLu29W2x-oSwLIifu10nbrX_ybLNIjg5m7CvO5EXSaUJaJkqFyHxxW7xoPBSlaq0YSRk0mfcq7g-FzxxnJUiQrY-rCp3uQdew2RkXEiAd_u6dg/s1600/ex08-0223.PNG" /></div>
<div style="text-align: left;">
Be sure to follow the correct order of operations.<br />
</div>
<b>Simplify.</b><br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-S_sRA8Xb745MrEby62XL_VBV0kunQXWo6Xh0lMCz27CGQ5okz8T2oiO9fzOeHrULDkRaJOTFRAhg6OEMnxByXYrEUUBDHwdJy69S7tXni9yLSufethhU_dIcLTRoIrCQWm8HNHhzjao/s1600/ex08-0224.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_bUCOi1ZHl6x96rxKiLREizVLlQxZSukYY_HAjwki89pRX1kozmVi71NKVu0_NXuY7nSGZZhi-UQ9OMa96b1eUWBLVg2ShJO5vZUoNeD6NR68fc03W3mGdpi3zYnaY3Mrrm2GOwzAQqc/s1600/ex08-0225.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhx8qsIeyS7yR9Vwo-1HUHjJqxHyOkFzcuihCI1ETcTQhD77hE7rHYSfDUh65SQjVBWDeqXh0ZZuUGgd7N_Z7Tlb8ItiluNEXcOGZ2U1tKGVBrY_3A5vMWF0cpi3wDxiBbu8faCtlfWTXw/s1600/ex08-0226.PNG" /></div>
Unsimplified rational expressions may look as if they have no like terms, but first try simplifying and then check for like terms.<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgp89mnnUYQCV3MycNwbJHsssgvYrdomSdiPT88X3L_e-9BgzeZHM3MLEEgXfZ5zItM3RhyphenhyphenlbcW5XoGI-s5mwoZXOgB8HfPAuoz7TJ6SVHI0GO3NcMwDCwYajX4ekGsRVAcP_ZsUeCckx4/s1600/ex08-0227.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKUr9PJprO98McbHjDznJ4GNBKZrK8vzQr8bSf2u5f8hHTIwOFK6OzL9ht042nUzJfoRUi8h53TLfl2QvXcw9_J61G3M1zd68EmHlyq1y_Kjmmdy8Jno8e75gTI79XCOYmLvJBB9-4qaY/s1600/ex08-0228.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiUykq25Qmo1nyT4LrLx7YNMm0ft37ITvuGnShBXiUhbDsP-rN7gMbBph23FSNOYZGjqPoj6MP6ECVhMRS-2IVpkXK2-KSjFHJi3oKYzeC-0Xg75BIn88piHG9_4xn1GceY0H9hS6V8Uyc/s1600/ex08-0229.PNG" /></div>
<div style="text-align: left;">
Simplifying is a challenge for many students. In particular, the numerical part is the source of confusion. It is much easier to deal with the prime factors of the number in a radical than it is with the number itself. Create a factor tree and determine the prime factorization of all numbers before simplifying radical expressions.</div>
<br />
<b>Simplify: </b><br />
<div style="text-align: center;">
<iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/O2z1ro8JZww" width="560"></iframe>
</div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEidMLDMNWYDVwPIL6gvlGoFz5WCBPsYHKoCv1D708yFS3utnl8JVTj9i6lU6FA9wzcUFN2b7pYpWB3eM_r1DTHgQGH6Li8gRPrc4Qhhlh-RmdBgkXv624OMDmCxreRN7UUCeILqqLOlPR8/s1600/ex08-0230a.PNG" /></div>
Begin by determining the prime factorizations of 108 and 864.<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-x2cPliwm5v2PVHlqdfxaJTDcqUZnMz4AF0hjzrTcKPd0wkDnaxy9s7SAaWFJmNj82dgtwmX-j268LMRqGyq-ggMGcpd2lq4u__J93TC_iZ_LZHQoaO7fLdtppBLiHnGxzkeaVA3EbKM/s1600/ex08-0230b.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgy9ek4iAsqSwHCDFnpEAKUEJRd-GdyimEa6FaOXqMhGK7g25VuOdbqkyuBZA1Mhic5w84A57tBOyWR7tJiUek0gYYcBvACXggLpgqEwGNvKQpYfrX-KIKKmKF8aBvsHf3FulvibsyKvZo/s1600/ex08-0230c.PNG" /></div>
Substitute the prime factorization in and then simplify.<br />
<div style="text-align: center;">
<span style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhWwMMSUYKj5_1JOna0Ti7_jrKzD9tZc2StvMw8G0eLZPECUq84cRcO-ie-_ta6dGr9Numz5ev5P5BkFeKiicVAHySWIjytHBbbNDZTz_T2H_nC3r_a-LzQ76l9qPD4h3z-zaPThDmXSS0/s1600/ex08-0230d.PNG" /></span><br />
<br />
<br />
<br /></div>
<b>Simplify.</b><br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgL8-79Lnsqqn5PoX9xxeA3banniSOVGHIxN04si16LUBYHUOEwkYABdTiCsoGvNVKD08Px_vJt-EgMOTKXLR1PEFU3rq8BBFB4CksPIWdvNDi1poZqfX5DhxBr0rJLYbJ7qPu-VYpgFGo/s1600/ex08-0231.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhR28FK4KGsIr6KnMP8N65jTtcrSCnwbFfqCetbIl2IDfG30JIBeZVqF-qb8VBnEklcof_XEXCMWpq3s4Q8d5xOKtdZGGS3nIqMBmThyq29X1e-uWgDGvWOki_uHa9VYypfc3tQ4-nW3Tk/s1600/ex08-0232.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjiptvIwm1gNyNieUC1B6RPVFgqb1XR5XcWoMRbz_9MGMTnQW_jDyV206IRO6TUft_biDYpyjumcbtTETRo9MulhZ2QVLJ4ZY4IVH4_hpywGXpDHeP87QoTuuvUHW-aByGmRy1UNlu4GfU/s1600/ex08-0233.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhCHW0JOa1vRLpzZpSsnvI0Pm16QZvlTz7cLF8rQf0QyGa1KkSDbRWqY7-IHeNb0NW91vSMBppZP7Y66laEN8QbJKMHV2CKWj5EjPQpgJ-o0cLftFByqAgikG2t1n3T1qpmPSYS2fyQ03o/s1600/ex08-0234.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEit80fhwnupDq_mZlvWtlTo2VpQpOgvFhKEGPp2Wg2Ph2JA8CbUuOcXuVfKrs2bhh4E8OTQiIuoONbjX64JUC93cUSjGYxo6s13CGBgF1ezlYqMCuiUh_3tdYdg41Wy1vxVfoGQu201__8/s1600/ex08-0235.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjRLYSR6dMyWQcQKl-3yyBd-5TB_JUTf0tB3e0rGQBmrYZwUdAvQ-RzJypcjXRdV-K96lswOcTlwRlfCXwHNSREcnfhCt-Cx0FSB8bAjkbukC9zDUcgOt8uB_poyfmfDlo81zTSwJPCmU0/s1600/ex08-0236.PNG" /></div>
<div style="text-align: left;">
<b><span style="color: red;">Video Examples on YouTube:</span></b></div>
<a href="https://youtu.be/wQDMPoAtVCA" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjK4NxTruWEoXvYBOfS5HrsnMaZ-cYYWZP20S3m_T1VQ8uC0hqBX_hatWymH9t3EV3386CcfT3fr5faJQrRAwWDsE0yzCpJe_oKpWu5VMf-M0LEKbnMfcV0WST5FtqfQBYJT4L4-qInYgg/s1600/vid-0807.png" /></a><br />
<a href="https://youtu.be/f-58AV9neSs" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjwhN9OU_wGGJBog-V9XcVgvr6PEEjX3Ukzrb5nnjANrzcygQQQ-vNmies0-L1YmgxmM3ZJHt2dOXxrAV6AaU26Dab7sTXA-AmVFAh5e4L_VvBBSq28cIAYYIAn2j6li36uJUgHixOIISQ/s1600/vid-0808.png" /></a><br />
<a href="https://youtu.be/qeaXrcybGdk" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhDjQt9vdoduEHwVfZtMJr9ZKyCQ7Tuzu_cGSTXa_nwmVShqIduacEfS9JeEJ4-Pu19VB_XShxlIp5QRoY9X_VoBaZFb064yu9gs6kLSmUlxe0JPKKd7j3TeY37V2XuiMBxQ95uxKvVmXg/s1600/vid-0809.png" /></a><br />
<a href="https://youtu.be/Al7V4AkErdQ" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjlSN9-atix5bWOZkh9Vr5RIGihCBHFLCqlx4LG0kcSREE7P6LXhpcnRHfzpNY3mRfBzWVo3uT00_F0ebmwdDmSo_Xi0hsDavcRYfrPH-gDJaWkMqjkIe7vxKM5_6miK37OD83d-Ao1X8w/s1600/vid-0810.png" /></a><br />
<a href="https://youtu.be/Q02WlLtm8fw" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgBIarjMigzfRZXLvdZRtAwAFt-s3sggDXzF2p2d5dN98cdSa2wwNUXVPSuWVL1AprXn4_A3FKM_lEo6zmEtPjz5f0INmPD60UF-xOo04gSs0Atc9MLBB-E1oJP-LrJuAW1asuYg4atrCw/s1600/vid-0811.png" /></a><br />
<a href="https://youtu.be/PxjMl2pROLY" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjxwM0tOMaK1eva-IdPqbXjvlAxggyjqVsncmF8j8a7cm51SGo8O7v3BelkpAoidP2o4rbYWq4_X-dqiLT4QcEhLL7aHVMw1uWYAf94yX94_ulMXUaF13SwK3Xxis4qKlYlE65bXqWHP5Y/s1600/vid-0812.png" /></a><br />
<a href="https://youtu.be/lC0ZOWzTm0Y" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgB3ska9LnqFlXh31cZvlwQTOrSMT-kaDhS22QHSm42_SJvYJeYwQiw7e745afe32B4dkTmdL5WxTyNYCkir-djoQbGYaFZKASZXn0hcpPJWniHMDmF8HHm2b65EA5L0qE_CcYq9zim46c/s1600/vid-0813.png" /></a><br />
<a href="https://youtu.be/O2z1ro8JZww" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-VLf-5Iur1290_ZdjKhezCECVq3d7ac-kIvKyKjBATHO6OxBvALniEcEUupfVEhTqsM8SV8k6t-4OjAiS5zCuqyqaVtUInvA2wEnMYFi4hUmNFLYeDGZ7JU9-vK_9KKQTNVCym2Ot9aY/s1600/vid-0814.png" /></a><br />
<a href="https://youtu.be/3a7g5SwgVJw" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh_5dnpKdkXXEKC3I88K_aGPcxH017yQ-7QRfQE0iwy25WJkHerG7wMj7HMZ4vln9affgrHJlnKgNa2oNd8nofF6UMbblg-J4GPOoonYAp52xQSo50yC8NBOH3FcRDBm4wbmsn0Kuufyvs/s1600/vid-0815.png" /></a><br />
<a href="https://youtu.be/M4hh-5mvSx0" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhTY-0c-kjKK0ed2vI0Y4OUUw-0xWcg4o8VBGnOaCcvv72mTWw3KtaMlSsJUhsY0my4SVtHqO24szx-WLZIhoCkNWGO275JwFWY4qSZ0pFhyEkzA6wpkZPsSGGF_uzyMoQ7Ih_WFlrW3xQ/s1600/vid-0816.png" /></a>Unknownnoreply@blogger.comtag:blogger.com,1999:blog-221233850938098964.post-77807597517156533642018-11-19T13:03:00.001-08:002020-09-03T14:32:44.139-07:00Simplifying Radical ExpressionsNow when working with square roots and variables we should be a bit careful. <br />
<br />
<div style="text-align: center;">
<span style="font-size: large;">Playlist: <a href="https://www.youtube.com/playlist?list=PLmN1jmOiJEf4hOOt9DsjxGV99vBDSMsgk" target="_blank">Steps for Simplifying Radical Expressions</a></span></div>
<br />
The variable could represent a positive or negative number so we must ensure that it is positive by making use of the absolute value.<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj1L9RklrBC0y_rvuAidS_CteHVMYojYMTgiQBXqEIHy44ryjuZPH6j6SkwqInn-45zSfvim-N94MVjYaqkU9CjD2aNZvLowfgz_DK9ICiIAzyz7VowjdBi1CtLmZman7DF81er8Tx0om0/s1600/ex08-0101a.PNG" /></div>
<div style="text-align: left;">
To avoid this technicality many textbooks state, at this point, that we assume all variables are positive. If not, use the absolute values as in the following problems.</div>
<b>Simplify. <i>(Assume variables could be negative.)</i></b><br />
<div style="text-align: center;">
<iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/on4bStap6fE" width="560"></iframe>
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhIcL2wwBwWBtRYd0jgfBY_fmhmFVMJgCZq-zEGMAsC0xQ53YOWWAXaenkY0AkqsdzlHXOYPxqEU3ytwOF_iMO-NCTaEU16E8arZ0I-JFcucATY26xitCd7dPqhbRBuDS5vilR2aJV9izM/s1600/ex08-0201.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvCEd_eVBoIvfmtwrT7eft4mT3cDtb9e2-r1m8-WsEWMiS9qHcF6YJfDNuoZnS3PcObaB1Zr_6BlpDHh0FtE0W8HY9ERk64nVDewEb_y3v-iSpwf05cGciLzJGAKSxBEvApoXo1G_UXPY/s1600/ex08-0202.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhxOw_fxtMPN39qSdlbeeXiB_dMcHgMmEhqQBdDCs8fFwH2oIv9P7trkX0KtkGLJdySKkCN3DNSoiVi0Rx8xZ8n6aKSdO2sZ4M2OC100UpeL3gKlKyjZ8kMTZCKGx6OQboAYywdth5iQHo/s1600/ex08-0203.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjyzm5mG2y8WCacJ7EWlUbf89t_m-KbaVhGkOoCjDq8s04KQ9MDRMqASSVVS3cohfe9WQOBo1WlVnSAW5IO5zunK0gu7uh-Qf2Br_RQNCT0LjvCGAr_DMc47HBzjlhtzxHqMF1kUotPYxk/s1600/ex08-0204.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiQCN9sXerwzTeUroZkb9HlqRpINKr6HuRAwSaczqYfabQSrBxhwRRygJs9P5Tr5fFGQUnzcSc61eNGPZUVBEOdb6m_CBciabVEwUqGa_Myj0hmQwI00BRlm_nEQ0W0muqoZpw91-gOf48/s1600/ex08-0205.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhj-KJV270qsqz0T6Of9bpqM5CJjNt2A9D99jgsIWAzgSJYZcsAqyWMsS7gUX__gdZSpSZSnX6b0N5qHG57ihq7AkGTlR2ItpiL2gJy2oYQUn0XIIQ7FN0nrUjJ6cj62Q0kvrzBGeik9Tw/s1600/ex08-0206.PNG" /></div>
To avoid many of the technicalities when working with nth roots we will assume, from this point on, that all the variables are positive.<br />
<br />
<b>Simplify.<i> (Assume all variables represent positive numbers.)</i></b><br />
<div style="text-align: center;">
<iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/gqSrCOA7BGY" width="560"></iframe>
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhRzL7rOibSTUfFO_7dMXcQWMo5ucE0Zwy5814HVvsgwEcEzcBIH21oxDFcLMOMXUOXLGImV69-X0IhMOKJIRR7SvciOHQlz7AgZ7QBxVkiHy1gPRS_hxM8-eUelFyNJM5pGTnS4Qo83BY/s1600/ex08-0207.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgK_nU4v8IiOgZiOvG-ibjAQ6oDcWT-ioAA8md2V8tyNK_wacsV7NL6s2nWkONVfAOEc6FvS1V0UzMup619Qrh38sPxrYu7Hl_Zj_023euyKdqrOEbW3b0Vx5Nus6cm5OE9laynv0jpR7s/s1600/ex08-0208.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhB1ZAWaFrOFf2oPi1m-WgE-1baRe0jVhmLiURc6vFu9zs0z-mUOC-15rwZ1i5uM8sXjT4ugi96MdT53MfjhhsJont4EKNs3pFBRqRj02_Tyb6AKoD-zJZBaOfHqyC3kBc70eCsPSsURBM/s1600/ex08-0209.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiiNImz6qrQr06iktq50H9SJwTBloNofZkmKoI2aQb6KjoYQyXt9_9StNxDfMkSV2m822kjle7srwfDvjN_bEPXc4-HPCus7oJg2fZgecBAJLpD-PjegiBfF4ILYAGeJEMLPXx9zaD_mbs/s1600/ex08-0210.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg9lPVbLP11lXjjD87v0HaMmRAPzYZzDAIAhVrevSaFCF3unbX530JccpeeC4PkKOvM454HQ1l9QuIBflEjEXfjSowujxVNo43iLpXabnMsZUle-uJ1d41OKzGnizoODPVd0_zzX8D9KwM/s1600/ex08-0211.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjjAqpwWCcYQIBM8LNPr2aQZgo30prX8tv2rrXnrJwvNWuZ1SZ0FBi5X6I_X5DAIwcRlg35zC_W1J334nb6NmvNpS95VubxvoOi767nKDd7nrcn_JptkW5MTtZ0JFQgUtiZiasZAkOXR84/s1600/ex08-0212.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgAAHQXZXxWke5Hlxk4UTESrYS9yAK4_lveOoLisV21drhiXZPnKLh4kh7OG81k8Ls3DSayRuzddaNaTxxZc_s-0KEbzgj7kvqmEmaH8NvTNfVX-BOKhu3ytnjtbFTUxxvVgs0TwgmfKro/s1600/ex08-0213.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj97jvn9ESzXnaBzDAupfPC_O_F63ph7eTrhP-T3Mq4un1fLcKxBKX6uDmKWma4or2QSRIQLxV0gVOeub0DQfR2gflasJ9GPqX0mmDS_p2brdUBnQaJ1zKTKPFw0Gk3DZiwDCcix6kTfqk/s1600/ex08-0214.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgXBCIKzT4w1bv6udQ5A2dTxyinzNklYUTst6PlsjULTA_Y0lZ1ioXI2cD4qb_C8Xsq5xQ_YEFwToGFPU4E70nHBeOJx-NpMRtw3JL_al_WyZXVLEGpMj2SPDKg5Uql3jMs2PrjT4TRaB8/s1600/ex08-0215.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhdhceB3kKgkovkbhUg5DY_kufmI2llwgtNeM61L9lZrSHmr71hvsSe0U1teCwKYCmuewZyn7YUXTLN-e18hGPvEym7qLzOlyyTbwEetM5z8d_Q7-Z7n2gg3BRDkexo1R9GHDsJi3goUTs/s1600/ex08-0216.PNG" /></div>
<div style="text-align: left;">
Students often try to simplify the previous problem. Be sure to understand what makes the following problems all different.</div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEghAsVldrjB-woFiKhb91WurQVzE2rsnkCjQjk0bIaKveerkFR5bsMxf3mXkLLyAyDdBoQseiMtnY358MLr7SJ1RyRKkw8mbANSTG1H4B2ZaMdQYlM9S6dFoj0GBjRMBeGdLeigv6ZpwMc/s1600/ex08-0217a.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgHVO335SxSzUWDVsRGPFgPEBkEI00YEx6LsnrIdbW9ruTAU8AoLwTXeUauLyta4fdyNS1XS-0G36YUBUFWyYrZ99a0PCDmw3H0l2ZeAkCb9b-oDye8ytlLDsJC8wibUsMWmEXb3sYt0Rw/s1600/ex08-0217b.PNG" /> </div>
<div style="text-align: left;">
The property:</div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgv5sCKW5vBQeaqohjehDf2iNAdZnBOZrQ9dc1j5YDzLWFmd8hpA_9CkQ6Z1-Yhb__32yLbyCOPMYNc8og1C12ciEqSBbyQgpDzgCzWovcnAECOoRZVRWFHXsE6TKe80-S410PRRcPJkRo/s1600/ex08-0217c.PNG" /></div>
says that we can simplify when the operation is multiplication. There is no corresponding property for addition or subtraction.<div><br /></div><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" frameborder="0" height="270" src="https://www.youtube.com/embed/yHnK34WzWRE" width="480"></iframe></div><br /><div><b><span style="color: red;">Video Examples on YouTube:</span></b><br />
<a href="https://youtu.be/on4bStap6fE" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgNhBK8reD3ha9vD38aRFwI0MJ3_gpYavNx8I3biDiXg0Uvh1ZsDnLvXWbkjD247KMkMhMsWBJMYUR0EQt2QGeu7RkV0Bc1zZJAKGy20AulXnH7ArP9uZWVay77QJwrt509NK9b71LH-fI/s1600/vid-0803.png" /></a><br />
<br />
<a href="https://youtu.be/g7df7MQDsi8" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh0x7eNE_GvKHhj9MDSLDa70DvZWPgWlV8Xt6y5K2waF3AkpHDvv5NGNM-LdSN-5MA0s5GhqRwKB3v-71mhdD6b5BBrtAjOhSP6BxLSQCG0979EjS08wGkcd3Mkk2NtBtLAl2WdjUiaNPw/s1600/vid-0804.png" /></a><br />
<br />
<a href="https://youtu.be/gqSrCOA7BGY" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtZSYkz9Ty7HhPZOo1BrFxX7iJSXt_ILkee2NfzylhyAVXxKaz55LlA_06VQNWUxlwAmZn_TP09_Nd0MnNB0hC_LnM0aY-IVdLcG0BwjEzKwz4chrN82DWOYGCPF_wCVYDd6qU1EV-qvY/s1600/vid-0805.png" /></a><br />
<br />
<a href="https://youtu.be/iVT3UvO7X2g" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiUDmayQUviBxq80mDEhlGXjDOyJ1CWemqqddsmelDi-kF2zJ7MakVfVeNX4vqD4b6mdyfGkTH5NO1h2H94gfnzkQf3G1rv38z_yLvGUr40YhTHuTwQF4zzoNY1s9BKZPiqBL5BTjZkKEs/s1600/vid-0806.png" /></a></div>Unknownnoreply@blogger.comtag:blogger.com,1999:blog-221233850938098964.post-72942729044125588482018-11-19T11:06:00.000-08:002019-09-26T10:48:59.829-07:00Solving Exponential EquationsIn this section, we will make use of what we have learned about exponential functions to solve equations.<br />
<br />
<div style="text-align: center;">
<a href="https://www.youtube.com/playlist?list=PLmN1jmOiJEf7sSYn6Z5BH5vIR-m_6nnMf" target="_blank"><span style="font-size: large;">Playlist on Solving Exponential Equations</span></a></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<span style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhuY53J4LlStGLrbQjpTNxoDF_4oFVkslQJf_YOpXYu3tjaNwhcf7eMvmwbebHvRQ7u_kUfKCzPJKRp9LWJN5yJLJfwRnDqlWMumZw3Ma0Y1Frsk5MbY7gyyfd7TV8qkYjTI3-73C_8sJk/s1600/def01.PNG" /></span></div>
Make use of the above property if you are able to express both sides of the equation in terms of the same base.<br />
<br />
<b>Step 1</b>: Express both sides in terms of the same base.<br />
<b>Step 2</b>: Equate the exponents.<br />
<b>Step 3</b>: Solve the resulting equation.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<span style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgg4A9XC6g9VZajyPDCTrtXLzZ4SNHOPtqkubwSEtDXp7SbqI0WlI0O_P6Un_GcH29IDBH0mALfovtSEIyrOC8bwiBU1dzHWadbkHIPaE_sGsb-YK5lUADqzwFfSCyyG_0y4iCQJMeLvkc/s1600/ex01.PNG" /></span></div>
<b>Solve</b>.<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgQ_0gy7HqQg5w-jiQaQYUhpHd_hLeohJLpmSZL160IDqWOtdYxogXBjTKVOpscmB-HTg9tIGAWCg1n9fbgRu-l0HeZC8SfvPGFly2IYqkKIVC5XDxg-NgsSI0uMa91dcbWuuixOnChwNE/s1600/ex02.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhS1KA3I5laf_Lwh1OgzlMHkN-V6p1P4e0sTa9x21dKxilV22KV6QQ2ygEEzhX8KmI5WWeXxanVMd7K_LVfaH88GlJ6Bcoibl8K_oydsoshkkRilZjtgnSVcemoD-biRvHjFcc09DMVi0E/s1600/ex03.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjKluexj-wqaP8Zu4ZsTWp3aZZQdwlz4x9K7__jjyA1o-_0LF7MadKKkDRjsxxBLUl58MLYAxZLHX1zfFQ8z9iN8DMDBqneY9b2P_n9ZBcIUjV_BbbKKo_5-ztc0UmnJEBIUhIaFaVOPwQ/s1600/ex04.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhBCDPZPKFJ-OM1hciveU_CPBhKp8g1qZZdumGQIs3QEb_fNo10KfBMfByL7lemdR2XgjwQFrIP_Z8eWw-an5z-IZKv4egvjCBcCUdnvnVAoS96B3oPfkCLcfDXJTC5QtxGdl_8G2e7N1I/s1600/ex05.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjpTxMaDTbHvaiulusur8c-v3X0qCZ-QrztxdgciE0Q7EzAamsLAy9BWlrZpmN8-6v0Xh-3peIxvhyphenhyphenDJMY_ORqFJOaVacTGB0g_F1KGdX0LbzXXVb16dErDMo6jXZfAO839no-Tyse_wF0/s1600/ex06.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhYS2SDBFjz4rewoYEIEgzU0i45-rJo271f15V4UkdX2r4Nh9OpeQ77cDkEsvGE0oV5yURKggEnPlENwqlIzTqFqT2RzM6SunMZGHsHCIfzCbL0gEuWJK3ECMHbzraHtoEqqKH1mWUkGVM/s1600/ex07.PNG" /></div>
<div style="text-align: left;">
It is not always the case that we will be able to express both sides of an equation in terms of the same base. For this reason we will make use of the following property.</div>
<div class="separator" style="clear: both; text-align: center;">
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Make use of the above property if you are unable to express both sides of the equation in terms of the same base.<br />
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<b>Step 1</b>: Isolate the exponential and then apply the logarithm to both sides.<br />
<b>Step 2</b>: Apply the power rule for logarithms and write the exponent as a factor of the base.<br />
<b>Step 3</b>: Solve the resulting equation.<br />
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<span style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhcwPoVIXB08jMqyPVZoG8RPZGV6SP-y77u717MrAsyYmcmJi7-PLSQ25UfKEUawxE0qb5GljiiG_CZ5gMFnazJWHyj1ZTKHavgQbHGFgiuzqIhh_ouC1HM-clHTHnxhDeBz8wvxVQoL9M/s1600/ex08.PNG" /></span></div>
<b>Solve</b>.<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi1VpbIus7MdpxFoDNMEVcMrQVstk25vGx4hyPvo9Z-MeS0L38Q9GFB2MPI1F9lEEfftqte9e4kfln28VkqUjnB6Yt5GfhdSYp_rRZKR6qjcMY6tmU4yfxRymsEVCHVX_-E7uKxA42hyphenhyphenlA/s1600/ex09.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi8KtpeKX9I51dRDK2FepHYki1fpEm5LKSuDjAhkDtcq2H1W_AQrVc59ibUG5HJPuknCZQisEpR5KhfaattSKU1NS3nwcpuLTImkBIVlxGFoTcIqz-PoO8i_kCMxe-dk_5pdCvkGnMXnN0/s1600/ex10.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYJC4yIposRjYm8BGq7vaFj5UFVlB88DiPZPunIwERvknsdx-eVNNedLU47Xrs0okjyYpdH0zcxaMLzTtlIer6AORE_0kdIKmLMSPvU5eltM3s8DHUHCqMmcF2khVs3V_pJ7PEG7dNzwc/s1600/ex11.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhHyIno7H_XjzESpps1FsjjaT0kZQegu5FV9SKtVfo2LuTU1TFr2U5h35g2oL97G7UZMqA3bEcP269TF-5Xh9BP_Uui2p7bchUQKZLgxczQYgdF7tMqt4TvbAc9RkLfoRWbyDzd2kE_rd4/s1600/ex12.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgSXi3-CjJgdWRFjqhhuU-XB2dNuK3zPsl7VrDDh-dzS0pahEGnAZY7q9Kl4O6Fpy8Pz8z61gptPFtkS_94mZNRLomNwgmZoUcsT1398eMojZ2DQWjYw_xJdzYTKlN7W-DyJpsNOUwu1QE/s1600/ex13.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgpIfy_OngWro_7anucj5sYsNldMUUh9SjL_T9NP60f5ZQcjEuHjjjlYCNmntMZ3xLYT0eQIcAL-qQzM7eT21hYf6Fqf5ENU-dkA6nodJyN5rkxbYdg2Snx0xVvhjcsdjDLKC3TrlYBFOE/s1600/ex14.PNG" /><span class="Apple-tab-span" style="white-space: pre;"> </span> </div>
When solving exponential equations and using the above process, the rule of thumb is to choose the common logarithm unless the equation involves the natural base <i>e</i>. We choose these because there is a button for them on the calculator. However, we could certainly choose any base that we wish; this is the basis for the derivation of the change of base formula.<br />
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<b><br /></b>
<span style="color: red;"><b>YouTube Videos</b>:</span><br />
<a href="https://youtu.be/leNS6vI8neI" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj1QZZ7bfygdNwJFnHBazASnYQ0c43vvUEuScyBZRZ7FJWCSLMVVy1X-RABUvAwyl6Q5Hb7gvub0tRE4bQ2Fyx70MnTbw6jfUajIJ_gg_pQZcR3JjxGEocCQHE9Xn3EbnV5YjpbhhPSJGM/s1600/vid01.png" /></a><br />
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<a href="https://youtu.be/3FRfLV0CM8Q" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEixkIQS447FULDSjzys4lGgMjJTd_YRT_6NhWEl2pVCY10Uqxye7vznhcsE09BmwfR3mAz6FnAciEzVTXj37FjWskQK2d5AmCvMclSrOfUcJflinMxRK8hs_xsaaL1wLViXgFHf0Cwg7c0/s1600/vid02.png" /></a><br />
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<a href="https://youtu.be/O_O6-youaEw" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEglWdGssrSikpnyS1iHqVvWGnAsqVfrKokRs-LeC4Ufjv724qXTnnIPLPaOGQLLaQaSFupSUA52nJh8NtdeIineCQyzysQ3g8HYQstTeomK4uUsjKG0o2WNKhM2IFDyZX311C7Fp5G-4r0/s1600/vid03.png" /></a><br />
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<a href="https://youtu.be/UikEWrwHZWs" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjefC65fnJHYQnWJDqn8FyQ9Ng3jtXuQM4876YwKpUU0h-2TSpvHHlUGYMhP89HAGInNuq7cVhLjuLT1sC6s_5NdcBm4D-5H9SBk5_TLZsCAOsBH4YMoYB9hAiPzMJCNMoSjHzOclkwLTQ/s1600/vid04.png" /></a><br />
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<a href="https://youtu.be/7rVPP7UQWUU" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtVaknjesiAsbb7aveZK_qlaJL8-HQT9C0VdvCHNDcVltC_cuT1c1GQLe1vwrH4_GvvkqHvc0254HlfeQgWy9FbFsG1QXRWOT6ttN2QwsZxqEoHBILT3d2j1psZCe3UcKwmQOBvRzNdBU/s1600/vid05.png" /></a><br />
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<a href="https://youtu.be/CobCw7IOyXw" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiC7H1d39GkPLPpkyLghlxhGqg10XyPl-vloLWdCU8F3mUWexTETDucJBjrnhSwa0Sj0necRnR6BYa0q5iMl9ueQqZZjMxKnnRjri-v1ljrzKtPrmvHTe6tXcK7Gyo_Le-BJNit8SnkGP4/s1600/vid06.png" /></a><br />
<br />
<a href="https://youtu.be/JErSiaKRARE" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEio6kvqx-tP5CA6iOvP2vBnT_eeEebZUMrraorfmxj-7DS64VI1cHns7EYqnWq-4-fQHLRhScw69eeacuCaoL0MhSYazISgzb0UEHKenJ3ASyOG0uV7NC1eRRFdy1_x5oBBla8ohY5Foq4/s1600/vid07.png" /></a><br />
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<a href="https://youtu.be/Lm8wV_zwK6c" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhDtN7oPFvnBI72QhBc5u5Ksul8g48Bz0janHSH1Kc0aRVlL3-UlUBqMoIWpNS5uBzi1DlysEmlCXa0RT_w8dO_5Vbfsv2QFM4e52EOFljdmeDInV3YmafH8_Cmnz-cvMhQUsA65smsgvw/s1600/vid08.png" /></a><br />
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<a href="https://youtu.be/9UWS0P52iSs" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgcNxkoc-sClCR1oycs1mWQLW90wDebQazXkEcEomRoiMEx0zSCAYkPZU9SA58LhiadFqBlCrVBivYFBPxRAaN5kdGrf2wVgLftIU2iki4oxRbkzSVPxOmRgRR2UET1gZslNXzVVvg0mmQ/s1600/vid09.png" /></a><br />
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<a href="https://youtu.be/ooAcEnaD5as" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj-74WwdjE9tnA1sH0rj5M9_Z8ZeJNjGeSrfpoGNnS978jcQFjbD8QGjFeP4QX9MouhxMZrquslcvaTAijZNGQXO3l-uP_Ebd_qX2adWFs_O30Hk7rTRsNQD-OCnocoSxRomm3vJJZp7LM/s1600/vid10.png" /></a>John Reddenhttp://www.blogger.com/profile/02231088988057526726noreply@blogger.comtag:blogger.com,1999:blog-221233850938098964.post-1290365884505843062018-11-17T19:01:00.000-08:002019-09-26T08:36:14.242-07:00Properties of the LogarithmThe following properties of the logarithm are derived from the rules of exponents.<br />
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<div style="text-align: center;">
<a href="https://www.youtube.com/playlist?list=PLmN1jmOiJEf40-X-SseQ7nxDUKCJ93eLN" target="_blank"><span style="font-size: large;">Playlist for Steps on Logarithms and Their Graphs</span></a></div>
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<div class="separator" style="clear: both; text-align: center;">
<span style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg5ZW1QDmkedpW8xjyC28AVI255kezR0nnESDNha1NZ0gBpWFyOgkY5Lm_NZA6YquwiZW1-b91YlnsgrwwetK6NaiVDH9zLfVskj3kpBfgV6tRG-iJW2Ww4M5w251Sz96ui3bv8_TNv-fI/s1600/illustration01.PNG" /></span></div>
The properties that follow below are derived from the fact that the logarithm is defined as the inverse of the corresponding exponential.<br />
<div class="separator" style="clear: both; text-align: center;">
<span style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEguend9BPrfu4C9y2pLAJD9JEA0E8ZwVym7-QBoRcc4AgZ-oF3w_amztFJkHNoavYlg2w4B2yUJjVGaJvhY2qm9k6S02-SJCYHROWOMIeWNhgI9N3Y0t3AX_xUL6RirehUSb3fR_9F49zs/s1600/illustration02.PNG" /></span></div>
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To familiarize ourselves with the properties we will first expand the following logarithms. (Assume all variables are positive.)<br />
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<b>Expand</b>.<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEibN0NfCWXeRAmfW8WcG5j3Yss6kr89dv-yzSCnf0U2rYfF6As448VzfRu0_kWE9fOlwaPZa7KyO0QaHaeFf5cZejIwCUmfKJy4SaS0xW3BZzZehhyphenhyphen2W4AhDTQ7FAcuAvYgt8hRFGhL4ts/s1600/ex01.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj6EOKwKJRqyPvNO8S-9zp7EPgCLIAEN3euUWbGwR5xBKZ0fXFpZRPvDkiH2iQ8Qr5Q4xXd5oxANb-mNZ3ZpA1u1ghO4ldfh4Y_dP-bT9QiWewEQlH8sLb8MK9iP0vNM0McawMduK15UqU/s1600/ex02.PNG" /></div>
Notice that there is no explicit property that allows us to work with <i>n</i>th roots within the argument of the logarithm. To simplify these, first change the root to a rational exponent then apply the power rule.<br />
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</div>
<div style="text-align: center;">
<span style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgOiHR2MHMv5U0xt-pc4x18otDwdNBPC3YbzTkcXJmUrz3z9gsC1VUkP8QeuP8idxodsQfuYNLuDromlhSYGdrmO_oVT-oSM9OYZ5GGogfBMrMFqaUcRO8dad7SDSwgjtWx_y4tRloCsnc/s1600/illustration03.PNG" /></span></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh0s2jfgH5NHuncN4Y2L-cuN3ZsG6Y8By0ifik5OPnSV5QjWDPYG1VZMCvkz-74Rbsr_dZ4EjzG-0sknpCuosSeWr1UanQzLT-5uIxKVjWxwkQ6xEzYprtt5SQGFr3J3un1Qx4KbMwvDck/s1600/ex03.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgI3LkY-mWR4F1B6nyTXJeRDm7HvXuASGXyBR5_K7p1w-YnSgXaN2SzlAoaHZajdbTsf1gc4IKCSS1t_ZjZUID5NLZQmhVppEatiWkV_19M5MUCJ8AIS-Wev4doAcn3atv-jHqPn8vvx98/s1600/ex04.PNG" /></div>
When expanding, notice that we must use the same base throughout the expression. For the next set of problems we will first use the properties to expand then substitute in the appropriate values as the last step.<br />
<b>Evaluate</b><br />
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<span style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhlLr9MX0TWNKgwCCeqdKNwFcq2fVjAOyZG971Fn8qcneIky4wBdbBaIUP6rpYa0W7UGg06bt7-X4sV_ML98wd0-AYaPMbAk3UTkaIucTjlljz-37N4cK35S5VffuF-713nZr0u37tYuso/s1600/ex05a.PNG" /></span></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEizeCobT6pTzeQvg1xtSLazPId1D3tQNRjRGU-MRDXgV4Fyznsmmu0Vpfz8UgpDzKLXRh3613QzR41rYJ-Nzd4gWl4Ktcy9HxPZ08Lpk8GwI-YyiZPm62LAahMqTGNGWdA4MmI0Kf_6EtA/s1600/ex05.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhLeoKLJbK49Uay-J6XO1i_9Dvyd52qvYnDMYbZJBklYWMop4dGTcw-NbNl5fjmhKyV6popg5CJQ4ytdDMdt1Exd3rH8rjkXjXcvzHa4oK33qHC4e2GzudzRceO9y6f1Dyu9xugmkAXO-s/s1600/ex06.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0BEHTbkTjdOBXC30cBkvgztmIp_T3s2R1o465aWPHrJ9-jyZQLL9rxL1bcnt5duCzFfTg4qI4qkB0NCWQGYzqqwq-TWpZInhfEWjU5Mho04h6yq9BH4Z1zr_dJHh3dtMHBeociwUT37M/s1600/ex07.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjfsGeFPN6bcahnHQTzzum3m3VhdZq0t_FLY7sqt2VZOEa4GS1tI-S6vy61w47Ln6555C9HxSnFROxIsD5qX0OPW4dcck0z2wEVyEiRrvEx2HNB6PJNGGzMYJB4JuAuUgiANhnncpmHxCE/s1600/ex08.PNG" /></div>
<div style="text-align: left;">
Expanding is useful for learning the rules and properties associated with logarithms but as it turns out, in practice, condensing down to a single logarithm is the skill that we really need to focus on.</div>
<div style="text-align: center;">
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<div style="text-align: left;">
<b>Rewrite as a single logarithm (condense)</b>.</div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjA2ec_2kgQGI8w3GZFnK_xMMoTxa4dmvnoZ-yIkixjOjTSZPVW_BGlkiCG1zSZclsN_CSOf69SVyqwMeY8FOLCALqcu5jxsBjwlnsAoGZhoMDmZOPso8fruP47_dB2GzrX8PeTiU7Pv_U/s1600/ex09.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgW3CnXdLgpsXPvtRBfzLqKrJ6RhcGBliittMXjM7_SBA7GFcdt_zsqWEvlHIOTmz7ffScCk63xaodm5c0BRGPbbkLT8pDyKbUefDjPmKEDGrXic_kQ9ntAr59ZutQ5wiiQCnu5TKsOPdI/s1600/ex10.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhq1D2glxXpubnHFt2b7ugJRRt703-ZKGlXbP_zvpIaqhVdriSxSLtFDbyXGg4dQrcuAxiU96VJHI5Wgxpl1yGcaeF7idpDoSHvAub-sjnzBUjjmu_8HgoXww3lUkmzJgLiJnMyWra54vM/s1600/ex11.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKBLKrjtk3szicT38npXcveodiV7Z9gcG0-GAsAzo-tE2B3YJ41xW0nsb7_ZwkhP6VqEgB0daHoATy4HAtZ9lnqlvktqyWDq5x2OoDxx5Xk5AZJTSsd9Q58mL8vVMnLWE21M1c_Ka_Nb0/s1600/ex12.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjhPlRX6dl7yhWw-rg_1tADqt7BZWv1Hq0Q1JbKjNjRkvigVr8isMizKDL9ll4r2Rzp8aZz3H4lWjxuUuJqIP1ebrxxVijiGL05YI2l7nboqcEIpKgYna0JQNyp3PznPywcrj3o-dtG2cU/s1600/ex13.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgOm_RaQ3LxXI-eK-ToLQvxJ7dlMT8eULW9CrCrSb5HE02RFhOZGFXcBJZ0L3oy6sPhW8OQ6goc2f9TX_n18_vGpJy-r47SV91sm5jHLzGJrWcLjy3iNzZUk-3B6tGlRrf2B6RthUQ-IR8/s1600/ex14.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEglR5IVUTTzQoWTlMGtfIsZjc3Tug21jeKyJMIF0MGs-0ywM6IafLkbvWRvKzrrUOFgRxFZK7qh6tejIQR4ZE71W9ogK0mLkoMRw4XOOzAxPJb8uQ2FR327aX6n292UACMqK2ZSPRipXjs/s1600/ex15.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhKDKImHuUmtbhOzQvoUkf7IXs5_GsuSKlKMILgk4tlR6XiBygid_zObajqgUDMOXZ4SNjMdARTxiILmUNSsJUBa9P9Qj-hrt3iHKazkq7hDmmFuetI1Yv0DtIv2_vXHX5DwWXjmc_NqWY/s1600/ex16.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhUszLrxuxJkCdMLmBYeZ0Z9Vic8yO_K1HrcOwrg0pe9MhRd3brx-J1Cm-psaASNjBOTcAnMPsFBklDgXkQy_lIc2b3VAwh6shE7F4nDIYna1lBtEs9Ypu2fV-vwJvv_bwPAvI3ZLjeq2E/s1600/ex17.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg7jMDhiMIHNZs-Wr-lflDiBgWLgScDkRDAgSgmd-lsEXG2IGex0R6nl9er0zjqmUhadRo6h0KeiGjWpJjlZDLCw6saPnFpMCwnaRlKmTXlpQ_K10Jl2hY-gv5bjjlN0UAQh0VTarTm5NY/s1600/ex18.PNG" /></div>
<b>Tip</b>: When simplifying these down to one logarithm use only one operation at a time and work from left to right. Combining or skipping steps usually leads to mistakes. Do not race, work slowly, and pay attention to notation.<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi381KmnMbKn-R-8f7_cQFVwwU4gXvv3FpNp7svbYz-jcUWjw6tMiECDdtMY7bG8De1bSkLiz_I2zlJJTvQZxCB11HOw9nATeTUngt9YFr88GEByjoA2_ZhxGt4I_yT_VAcxF0QoVBEbYA/s1600/ex19.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhAAsCMqzmUM6tZXEEzjZaeCBrvJZ0dfZPbIFigOn4j0fXvetov5_2BNcVvgg7cX06KUKXj5yi3jGLc83VdpOJW-Jr9i3ZoFNWPBUQFEM2d-6l1rU2vF_0W2FB7byzGgabeIbgrNd4UV5w/s1600/ex20.PNG" /><br />
<iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/CX3axffBAOI" width="560"></iframe></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_EWEFbh2XbswtiiJ2ipYewGIi8QAwo9dY8doJPM9oJj5nlGfxl3bNe1qG4wX7VdJj5la4981jthmEg0UwY6ZGgYpxxXrlKVNXHd-Q91GxmuzgE22f1KZwk7MLLqV4guTFG9xMmnE7AJ4/s1600/ex21.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqZhGfMEy678pJgXMqRjbcW8KoRjxneZk0EUsKDpdDYwpehMsnQc0fXGnKDuuN8HnvSDj50sj_YNviDT9yY01nrK7NQ3gTeMRsk5nTsj5QjIWH_EyXf_nxnIJNt9FDgAxhwsnJfJV_NHA/s1600/ex22.PNG" /></div>
Evaluate (Round to the nearest ten thousandths where appropriate).<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhbGxlUBBA8CGYsw1uA1yrIDyKUu6IdXXw91iLCq4kEJ3IAHrWUvQ1lp_4jQa83blN-wj1Eeu10M6if7m2l3_XdiqzYjm4pinR0jQ415zqmdvg7YEygdySXIR-yc_CeECAzN-WFUyRKYfo/s1600/ex23.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjbfeQl8hjjdSxK-Nt12peDNVLeoy4L2z20QzdIRKJ27VBJpF4Z9Z4aZpm_xpTQYkDXQQ4Ti_hPoKi6bFQhLUf9qaGpf9K4vYB-r8rxYwOLQYo3bqkz0Fl24eH7THZCNfwOvVDCL7Mz2N0/s1600/ex24.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiXgPv1UgTAaRX4QiBSDp3XvUrAfaVPjs5kmj29i-HFqEcCjxciqRJ0k8FVzXOWV-nxXe1QvPnd37rPK0lGwsa9VOlvJg29X8q0h4jdXfkNzLe-Y7hOSQyPZvC13vX673J7r-pUsw9kq5U/s1600/ex25.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhvyZdahqUlSDEiaO_0VcXk-HGAh21vPrdgCnnMtRCJ2EAi4RaPPnidvgR9354EWVdlt0LSqQfsTJhGySpMZYuC2Mlhf30dI3noGwFZ0eZvUa1pNxF71UWT5PH0_9WWb6kRzSVLFFhWzys/s1600/ex26.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhVNRr0rrzFxZR_EOyK8Lwr3lLwVO5mjKSIQRiGoi98_Copu8uwUoGVc5rL_wk67cWAtwXA-E93jb5xeWMmT1yweXbSKMQLODiUYS6yq34v-8rc0G8bzGyg5fBqNJh-cs7pcY2mXHQ6H8Y/s1600/ex27.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg5UmRzZsiqmwE4Ri5ofa4pOLqvyoCX911hEIF_FgyhaXw9_BaBrjrPfuwKGRiFDYNvqtnEt2KBkB6ZrIeISLZCkf1kDUvsb0j46w-KBsVYnuQp5tCNDWvwMcfGrBnKosPnf3TBR6JSZ1s/s1600/ex28.PNG" /></div>
<div style="text-align: left;">
<b>Simplify</b>.</div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjigP2lC7j2TJojKCl3HFx0czFZ5c2HHtKfjyM1hmr9sWJfAuPwwJliRjZPDy2KsbxgZkF_WWEcfXC4pAVZF25KMFOOQO_BvYmpJwvFEJdrYKuhyZ5caYt145JFmy87FobaR5hxkggP5-E/s1600/ex26.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjCpQ-aMaXiqxYrhwDIpzKDZaXkbkR6moA9StjY0ukRnO4sQ7uFDhcjFwYLhteHuymT68sIYaHGaCG3CcbyfSbpBS7Zwi9giLGCs8KP1pzRSvPCzUazVOWBEIPlA2gId3S4KmDHjc4cDhw/s1600/ex27.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj7me34IGmc_sLlA-ZyLb0V_72UmdWoBXGe-KL6B_4lNHymERdsUV8N8LKAM_5tSN8FtTsrUcoNzEGhGr0QbiNQ74Rz35zrfRLqxEj4M-aUUexPECn0RGFbYtxo141Mc3bKfAg3LkGKDWk/s1600/ex28.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg8i1hmE9GIvx-53ggP5PniMRQJEwkXjtnwO4H9PHPh1hUqbhIqp7nz_mz-ygwd7dWqSIVPDY_dy6u2t9j4DFsm74qJS6VT8ZznK6QGBZBhxnUFJzHJdVwkhNe5yTrS0TOnLOMFK3e-eXM/s1600/ex29.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg8TgdaJYxvXvuTGP3XYBSwgxcaofu11Nq17n0Vjt2USJaiDS7k_2699YdqVhkMGBMe1A2dHZ3JU_wglR-mT7YshmpOHaxrwVDhyphenhyphenVEx6w6-b-h6hNl0OuxmvxpNNFZ0HE6e8q1c7XRpElI/s1600/ex30.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiKxXvstdrC5l73lugxUIp8bKp3HyrgVQi7iRuk8DqWsG9X5hC4fbyR2gbbfrKFq8NM7TfDd6Oeudx9zl3xNReSPezPA4EmN-ogLG7rxU4qIS1b5sVTExwNtvrqBN2SahsSWeqqYRn6yFc/s1600/ex31.PNG" /></div>
<b><br /></b>
<span style="color: red;"><b>YouTube Videos</b>:</span><br />
<a href="https://youtu.be/KDJIfvtaVWw" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjpxdJwXetxInn8Le7Uw0x9tjF6loLpYFMA4eY7RQr5D2sflIMHIbk0ywqNlUVT02afD2AMNlNUXeMqjKDauIukY4nLSoqusdZL2O2GCBK3xAh9CjpqCrjtFFfZjvNKrYkqJUfvdqOyW5Y/s1600/vid01.png" /></a><br />
<br />
<a href="https://youtu.be/taMInxVi7Cs" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvhigcX2Yikc1uQlLGMUVO8JZNRpYxd8eeOvGi3sKCt0LUtSUrZ4B4-7CDX2SbxSe5FuRVDdc7XMnMbJjpuKVmEMIyMt_V76Xu3L0kNWsHcBREK_VfA7DKicJYX0WC1J8DLPEGOhJwEQg/s1600/vid02.png" /></a><br />
<br />
<a href="https://youtu.be/IiJyQ1-Mw1Q" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgpWynLxE973ku9plj8GqGUoM5PDg6n3gyaPrkGOjkPHRgvPzsN1iCmz7svqn7ysxv1jUOZr4LdHxffVuQdGuvuMBQS46yIzC94tE9J1syfqttbwahuM_sgs5GhCLErh_Sz2j6IRHor068/s1600/vid03.png" /></a><br />
<br />
<a href="https://youtu.be/JnQ0xvZ5uko" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhEasgUT5jwtstsTnZLED97mWeP5T4O3BL8qitOM9A9R7BVvdB5SYelRqgbPCcRIqWA6nV3yOi7x25Krs-keFXAwmZn3-1bdkDYGEIGPfvBOs0C8xErT3QC-3x25LawQkrsULo4uJoQXwg/s1600/vid04.png" /></a><br />
<br />
<a href="https://youtu.be/mWJ4-SO_QBU" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi8imnVTjLpG0iMTT1_hX_Wu_iyniW5VzYBExYiOMmq1jiWbWc19PRJTBoFetNmJ_yKmUPYnJP1EppHuDkPocYdKEhXFVTRg9aRsubaks3MrCJfYs7OaH64yROmqgSSar-xkx_zrWlQta0/s1600/vid05.png" /></a><br />
<br />
<a href="https://youtu.be/WIUmjCWdJBw" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhTaSTOHyrILTD361hEOQzt3BzCVi5ckhrCYg1ylA_1iNcqAfb_AJ7ORdttP3-YFs8pOj8gg_MAraVj1vJB7l2Mb0pckk8iWMauuo5Y_EtWTd3oGD9H-mOOPusCUeh27CDs9F-T0c5KlLU/s1600/vid07.png" /></a><br />
<br />
<a href="https://youtu.be/kt3ErKnMps4" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhYqRpjeUsc2Qj3lTwe-pYfm-ApZHdcSlN6XY5CYIhRAoh2wlORAyNyNsGtGF0PVxQ20ioDSicc2QokqGrGLRZ_DHQcf497wyyqN4DgBKTZ0DvWFHxmGPZA7ayDTXDtVrGNOb9MYMrmsVI/s1600/vid08.png" /></a><br />
<br />
<a href="https://youtu.be/d1CnNzbzVlU" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhpjl-Jvm-sThaEjk0sGKu8I5cMhUCC2oaovC-W_s3tKQMJoa7Me0xy5_UqZBq0yeo3qoiUB1hwMgZFbu4PanWksbDULHHl7UXsGnrgZVyCZ_Hnt2jGZ8Xu7m2kPLUx7bQKTO1TcQx6bs0/s1600/vid09.png" /></a><br />
<br />
<a href="https://youtu.be/CX3axffBAOI" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi8v66Y9KNVbT4BYKXVkv08-PMZgqbF8OcULXS5nDr357h2Xhvty6YxDsibQtGRXwZblh_2LI5iRQW1yuy5DfFc0Erz_IqZtLbE8k9O9Mgkf7C4-wKliqb3u6L8hIuMp2RM2KeQ88eSHxw/s1600/vid10.png" /></a>
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<br />
<div style="text-align: center;">
<span style="font-size: large;">YouTube Playlist: <a href="https://www.youtube.com/playlist?list=PLmN1jmOiJEf40-X-SseQ7nxDUKCJ93eLN" target="_blank">Logarithms and their Graphs</a></span></div>
<br />
This formula allows us to calculate any logarithm given only two bases on our calculator.<br />
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<span style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgOZ4UUAfL3nGHrn1FoTHW-tCa8GoxyO-b5xMamECrwohFUFro_aLi2Dbczy4U7kje0x4njlWLG2-S-CwBzDaHOONJZOBZ2JtAf4NJx4aiqz797DmkBlDJp-xiawUTgCtoCG-sX8Pte3Z0/s1600/ChangeBase.PNG" /></span></div>
Apply the change of base formula as follows.<br />
<div class="separator" style="clear: both; text-align: center;">
<span style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgSszAMSRz8eH9M1GfGRamVjSRvrG2KYCwY-IDy8SogBj54x1fOIZi93vGxnWTub7xmch7pCvq7NPgXfAJw5HI9kKFrh6hsxNbe6nKeVuLRMKxxtxut6XTRFjoGv7sHHHkRH7P3n-0D-aY/s1600/changeImage02.PNG" /></span></div>
We could choose any base. However, it is wise to choose base <i>e</i> or 10 because there is a button for them on most calculators.<br />
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<b>Evaluate using a calculator and round off to the nearest thousandth</b>.<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjXlir4G1AJfv1txY3JSXQWd5ONU5t148QX4GTowbefLIL3p2n2qrfPw_3Q_WYGZ_8MCTiUlw3TAgp56pzlA2U1Cc2as3k8kXwo0ThqxtAlwPSkMwS2lA47WTv0-Ft2rDxESVzYF1HtxqE/s1600/changeImageEx01.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiFhuATzgciXrtK-QJfuoMTW0NDimFzrgirmC8Pr8NTSUuyp5J3yiw_KTBbZYmpMQYaKknZA-jNVks0UOGU8d6W0ooNk3r9bndY9LkiiD66ccB6OnedplCxSGzUKFNhUHWa8Vlaglmpyc8/s1600/changeImageEx02.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgnSOGtHCygHlPK3gbTlL40aWjpkbi3oHD4MiFDg9hMpeJ8M6s2M9GWsVreawfnDCFqk2jQ_NBzVx4m56kxX8-j8XrwZG0dXeUVzT2UhUkU0yLjbKsKF8t8wPKKBLVyA8Dxob63NjhsGFQ/s1600/changeImageEx03.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEivbgXwqKsJJlaoAT7uHQunY304_8UdqC-hK54kXDkTBNyy-RTCFoLM8UqMWekCWnvbsquQLXwhhNY3cj7ClhZhYQxN7jnD5WSqv6aIWrDs1D7VGzqYSp-n9Q-uPJ6L7UvZOn-UFS-2hhg/s1600/changeImageEx04.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9cfKZ8uKAdKETFrwtEJL30QE3-4N3TweqUdoYCbw6GBfx2m1HmlzVFt75YHAjqRIC_iXdZJsIdHnqwpwoB-NTIHls72FcVgDwRPrK3R4R46Fc98CWt_nmlM674Ck9XjyJ4ju5s0mfUD4/s1600/changeImageEx05.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgF5ALSf0v5bfazp7BGA_v_UHfQAIwlbHrHnJGdL2zEKzVPyQpcHWUk93-tkhBWtXZmeivUXAe46ZmYa5OKS4X3gU2AKt9tSLCqpxMWT6M7FLGEB2JUkqkzbV8YIpwQREKrXRMSxEggG0E/s1600/changeImageEx06.PNG" /></div>
<div style="text-align: left;">
<b><br /></b>
<span style="color: red;"><b>YouTube Video</b>:</span></div>
<div style="text-align: left;">
<a href="https://youtu.be/4V8qqs4-eGU" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhpiF1IWspC0Tm34Qn3LkMW-XuDF9Kakuo2UMcwjfSz8Lnm5iY6NIo7M1f5dHrz6UAmVhSFysp8voSJrXVN1QjTLJln-GfJFudZWrhgE1ocf_ydbcYBupeDTVU6pqPNG3M3PovHKX0X91Y/s1600/vid07.png" /></a></div>
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<br />
In the last section we made note of the fact that exponential functions pass the horizontal line test and thus have an inverse.<br />
<br />
<div style="text-align: center;">
<span style="font-size: large;">YouTube Playlist: <a href="https://www.youtube.com/playlist?list=PLmN1jmOiJEf40-X-SseQ7nxDUKCJ93eLN" target="_blank">Logarithms and their Graphs</a></span></div>
<br />
Using the steps for finding the inverse we obtain:<br />
<div class="separator" style="clear: both; text-align: center;">
<span style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhj95z5wlgR6eTFbxRjxa0i_sd2dPCOgiRFrEbpQgAgrdFa47UvGepNdIwnXgOnayssthkOYR3dk54cpZ1BybDpJ9fPH67D2ybiGpbpdKawHHS8ll_NRyde4y8SDlsoFjEQfqwLVsQaons/s1600/logIllustration01.PNG" /></span></div>
At this point there is no way to solve for <i>y</i>. Therefore we make the following definition:<br />
<div class="separator" style="clear: both; text-align: center;">
<span style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgxz_reKoS4QWYdVox4ELwbioK0v9fbqxfBIsm2wpCqQf4Tt1Reu7l4PCwjM-QQnRqw8aXW2VJrT-pHH7IC_H6cgy7IzdaZWebDDAVu5G7tOuMY4r0TDh7BfuRbcuDi20uiuVB_tuZlo_Y/s1600/logIllustration02.PNG" /></span></div>
Here are some examples of logarithmic facts and their equivalent in exponential form.<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg46eWtdWdzOX1oD477BYCiawGcEEqxrgtmBiF9dW4vzfdxVXRxL37X7vNuMyGJ2o5II7BwAt_pn8Ck-co3FM4bxUrnnnZY62gk0eIcdlGtkUt3gPGpxRDqDgrCJbmONZTnhhQS7Cq7aTQ/s1600/logIllustration03.PNG" /> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRm5ENCqq08_qsKpeEfEV89V2wTLZXwP9-ngC84-MaCOWoimJbQb_WnbF9LwFGQis2YJJ0QRHeQtRyHfmYi8At152IuuPPRt26owUxReWPLeuSvp2cEpKrbiV7Grz4E4srEUHsK_0tvj4/s1600/logIllustration04.PNG" /></div>
Use this definition to rewrite the following in logarithmic form.<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhnS3dy75_A7GG0N2twmn_Wmn8GWrHk9JaG6T4mGjswl2vBM52XB0S3XIo79h2rGM1dyUNdrVriU6MiO78Auy5FIkOTMpJkx_iNhIO3HYoFIzvjtnBCCVEKDAFrIGTyFp_xD2rYfYXhzBo/s1600/logEx01.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgAoQYVpjqaKo3m1tm37-08CU3tqHv4M_qjMqS4p2jhsJib9sye9I6kHIIOawvLg_ZLZLOQOZw_T5O6nYvHRg4tln6JCjhG11Y2UzMZyS-nyq4E4BaP9Gn11oHJTkeu1MkuWUkty6WW-zE/s1600/logEx02.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgiOPcjXdr7wydPNRi_O0oz2BE88vBlO3TstnsSgxLO105yyhDlll9sCEEaXtPBPNOyR8HeLCnDBVmdgfnc5bfMcWBWEMqEXnD2J05uHtpKBt2dS2B-Ikhxf9RRT_0H3dXiRZSZ1pzR_pk/s1600/logEx03.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgS88uoStky571-gsSD4O0VeEFMFhtATKZ5KALjI8H5c4jTVn321ytib3Mk_JJ6FTIzBEjCxZo-vOBU2IlaG5VImO7DwEbYrDNkqWa_x6pwFS92o9rFhU7C_ChJ7VHqeMxcG7uheE202EE/s1600/logEx04.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhRCbTirN_QnNJxo8VugWpyQ9iUkrXz86gZAyFQLicEAGLl9kd0eDgRfZoWjOOUCtG8ZUuwGl7MVLqzn14QEUn2fIPTCvOCIqERiWY8JF3VZ8tIsOCZK1GrSQk3pmfV5SffGir_UgrdE4k/s1600/logEx05.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiIH4FNktJkl0uyVTbKrdbjMTiAnxrrUvy7q6TwaGb4CCm-BToPpFRW9AXnDMFPuI_0M-GZRUm4irI3fWXSQgBrAE_1xLledk7paV70zeMSGG1lHHCDy3BzuiQV-2aDKH7hElFetxX8h4s/s1600/logEx06.PNG" /><br />
<iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/Uyd3mBofcX0" width="560"></iframe></div>
<div style="text-align: left;">
Use the proper terminology when reading logarithms,</div>
<div class="separator" style="clear: both; text-align: center;">
<span style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjoC6sQgIiiODrRX9P2bXla2Gicftd__73BZzD_mbJAwu7WBBMXzjP4g-WMpUvTwxOZB57ls53T6dUPfrXZ4Jc9NiWA8_djJeS6Hkfyf-H91y_pRSNR1hFwimFNv0AhO_Zwz5iCwwXrYAc/s1600/logIllustration05.PNG" /></span></div>
<div style="text-align: left;">
reads “log base 5 of 125 is 3.” If given</div>
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<span style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjzYZiP7HSRQtOfFGEUBqmrIEJ9XPg1WFTYFWWFg6ucxq2r7KP3cf37_xh-mKFkXwt_z09iRTo02w7hI0KYSOG4AF5tnKWYmSY_40vzGDWPJm_z-9NAP_O0Cw9ELFt3UZ4-fZriJt7vWrQ/s1600/logIllustration06.PNG" /></span></div>
<div style="text-align: left;">
then <i>x</i> is called the argument of the logarithm. Also, notice that <i>y</i> is an exponent so the logarithm is actually an exponent, this will be important later.</div>
<div style="text-align: left;">
<br /></div>
<div style="text-align: left;">
<b>Evaluate without using a calculator</b>.</div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhqu1e0wKl6my9mA52PmgufKFaVOh4twUTFuh7t6vySWFkQ1ZGwDiwui1G2InxhaYX_tOZpgn78GSyBqib3kQtRdtS4C6NLGB0achkx90x75OkjudBfY8h2ix4iEY2xhWOxWolVvcnN454/s1600/ex01.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhHIcBla8BDyeSj3J9mQ2vP3t2-1vHPBDI733bb-ptYn0C1HeCLBzd85QsRusqGD2vEWOXewFN-1K2HCU63dqignSUy-WgN5A-9ti4-Nsma_UtE2b0smFBGCgDgOQcZ75CFCScb8VgXTbY/s1600/ex02.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhaBxY-ZQAJ7T2X2uTN3XxeL3oNKZcPMoF4z8bJsUu6ZLAJ_H6mOQusfZjuWF2aNdweZEv20Hb7Jfn1EflHCzWaMhJbjjO7VbTMy9n8cWhmt70q3wX3dmfjmHDDZ2kgbL-V976xxyn1oTI/s1600/ex03.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg8f2tN3zNfLiMprUwsJBGTmkb3heMMbjcE75m_nP-4O7E49yPTJcoiXKoS0aJ4y9HMwBU53eZRUjzRlS96K0BicZrzoeN9tgtAIaikx1K1KktKySWlaB0OPI3BgJocET6aY0l9qTKy1tY/s1600/ex04.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj0-4zdGW0RZ57nYsD0TmbLmJLHfb4RdSxXh39jg2JF6IctSRG9Uu12H54XF8QhF7XpwiqP03qJ5r1yI-87cyJxRqT9Q6QDiwhweDm4Per01XWH_UDfOplUVlBtNysSUcXdWfqi1NzxX18/s1600/ex05.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEipnUsNFbHnn_KxYgIjXHtRoqXkXDkYXkCeMOZUdBMIt8wP4N7l6TQ8nHvFJN4bjvH9BSq75xJaWUz5C0NIYZPtgs4_6sjvJw6OhMUp1zb8VIJYorDp0ohFfwZxSadhgkgNyTR23JHs-1E/s1600/ex06.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg93cGb1GMd2EcReyiUbsFzCqD2v5-ptUwy7MBR_wSKOoSOdhHr8omBCx6IvfnkS9siwdLs8qnMSrvRovCbLUBABYV986EGbtSaC554iIinAYJSIk8LRlL7V-1qPiCEN2-koNxt3X9Uwjk/s1600/ex07.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgoMS8KGgtZfP4qNABz0FbPe3VJeOhoYTq_TPGng91JeFAlM__80_qW5HW-52gHth5KdiGjDCAdXuLkxQHp6MIs2kZk6LWX5cf_Lm6NQDKZRg0cdfxMQsEWn-cLrtOyZsGwSZ-A9M825-U/s1600/ex08.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiWIqXgpnhOJhJYxVHsmcIZgQYT6QKnufmo5tBElExopHn8umndxf90bC9zc0D_6LKlDSFqNtWjRd0eJhTgAoFCyvMvJvGhNtV5_g2LK5ITmgAoaw1UyUSy3fE9Wt_zVNn9V_shDsmehtQ/s1600/ex09.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiddvupLLbBMbbbo-SOzbgfYWovDHjLke9vnZPhld2XH00b7wRFPXPpvQKIlxmLy4LyaDGfTAKSJSJwWsKk113fzlkiPj3qLOcE2nqXQ8EGcJpWId-VDgM1RktazxuM8iOdmKzgpNrhtYA/s1600/ex10.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhqY3TWYI1sp4UWa10FASqp4MNbIJ_FBEdWv1y3eOWYDKjYoH-BNIeZFIa6VGYQimnzXysVjeJnumiSYz9kZ2J5dNS6dcIQtJkViiGikcFshspHU6UXIIZNbuiW4ePkZwEQ0LClAC5HJDA/s1600/ex11.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjPa0WSiOYU1XkhwDVBMYGWLyCgk2fWXYHlXJFpsud2HxBomcBEAb3EwGhFTEwG1LIcR5cEYwev3gHsIO4D95hoPOFlpKF7cqYHj3IiLzVHB9sYnPx_0pWdyZZJJglWqi-81fUQI0F7jMU/s1600/ex12.PNG" /></div>
<div style="text-align: left;">
<br />
<div style="text-align: center;">
<b>Instructional Video:</b> Introduction to Logarithms<br />
<iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/QR-gleX9rr0" width="560"></iframe></div>
<br />
The base <i>b</i> can be any real number greater than zero but not including one - 10 and the natural base <i>e</i> are used often.</div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQvkYhyC1uAueVdYz1lw00h5pmIzVEFSp2Q1hpGo14WumpC2vqxyCbA7b7SF-vUSC2WShj-peeDa3TAodP5Y_NAT6VbiPhxITLLPnMkmTKEil-nC1g87bRrat1PeUkusyScXV6_Z4HxLM/s1600/logIllustration07.PNG" /> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiMNRmBHhv4O3jRTqwUISi1YLGEqXMqzO7QffMZsu20m6JF2NmvpiKlHwLwT-mXiQF4PryLJclhZ721uAmkDKkd4KJyXgQiPUivc_hHk6t7N9GHQbWfGMVvQ2yn2jGGPzna3VXrqaCMihk/s1600/logIllustration08.PNG" /></div>
<div style="text-align: left;">
A logarithm without a base is interpreted as the common logarithm.</div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiyl_N-xpQN7EJ9L2HDbKU8QtbfYlxVrRELfA8kryJqSP_P2nrLT4Rsw_lxiXw4Qv8JZzIwkJFYrSSBxxNPMcCGuRXwlE4V7PiEa557ihk2QqfoykTBpRiPirPnkkosY3QNXechj37IrEs/s1600/ex13.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjYyHz3QccDf0SF_Ja7Cbol-nmBkOEnNod3GTC5ofJHDw3L_3jI_c4ShM1nO6a69b9YBtBC0pckQrgv_kjw1evkdfeostkn7GHl9s_R7odt-AYwUzt2xgJbY9EETRs8gtabuvrd3YLfNQE/s1600/ex14.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjuAaOf_EOyJDIclvhYp5SG9f8-sYXuc5ZnJEkFR_Cf4QBbVbV_M8zn4hiJ1sJ-wSRGRaAct0tCl0hTZnnjhyphenhyphenaaoHMiXKpLsU6xO5inul4EBrMP_ABEMr_763my4Iog7NK-DkBIjDTjey4/s1600/ex15.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh10TRl1ERiQygABlSJb8j2Obi68T3XXoZbHA0s2NRO7G7Jh1zsPPHVKFTDVyZGXxKoyZpwGHNkiiq-dfESHa7sebmhuvssefgbMaMy2rmonKLzq6wby6ejwGhqOqASSEvopOqow6QdIOU/s1600/ex16.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEid9fwWBbNu96phyVvtTwFbxjWF5AQCdm0tcHnd5RCE54VzyB1gOwgXupDrOb-ul6WbBYA1hn5p3fjGSbg-EiIq08dBih-wzu3ymxQgoN5wYmWLheCsoFzyVxYYYw2KW5YL8f6ZiqziOcU/s1600/ex17.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEixie_WzxtjGnKqMm8a_baDKe1eCGuThzXnGRXnPjImEQLVV_p0WyA-00c0ZmP2J3_p3Phu-KIkWmavL27PyI1SsMhv6z8qDK8piIG-Ee1GQeci4iz0jWO3TBVdvibJuvCQQIMIuY-LeWs/s1600/ex18.PNG" /><br />
<iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/Wea7sq6LoRM" width="560"></iframe></div>
Often the logarithms do not work out so nicely and we will need to use a calculator to evaluate them.<br />
<div class="separator" style="clear: both; text-align: center;">
<span style="margin-left: 1em; margin-right: 1em;"><img border="0" height="107" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiEeH4cUeBwD3GpatsC5JqynPTFgTr5hzSQVHKuQvKQDAOfj2xjLxkl-JYTG-irOZY4t6rd7DYR68mJC3t4HDx-Lq7M6KxAdqPRMweQRsiq5Vx6RXbO5DltBCBYVR6gn5vf-ilY-h0AMUw/s1600/logIllustration09.PNG" width="320" /></span></div>
Here are the steps for using a TI-30x calculator. Other scientific calculators are similar.<br />
<div class="separator" style="clear: both; text-align: center;">
<span style="margin-left: 1em; margin-right: 1em;"><img border="0" height="69" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhCHCkuXyd9jx9szrWF__zJbH8FhJsU1cJrmBCZV5vi7iIB-yIGtbyfyJ7aZRqtq92IRuE5SbnkiHhkBbs87_OqMqoj8bKRw3_wPp-J0PpWA9euol6lS0MCFxYigMUliLD0BUXgWwWT__c/s1600/logIllustration10.PNG" width="320" /></span></div>
So if we round off the nearest thousandth we have<br />
<div class="separator" style="clear: both; text-align: center;">
<span style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh8mnqCsrq5WxmSDbynrYPD8gSUtoQOA8LfK3hZ0Kwesd1BMXw7XU3a-J6ETQEMCULvHldiVLzBQr3GNU-aAVmy9nKq-ysLsFKDr0uPobVpJQIq0ZBap7zUthS2SBTGj0sOYYY7DoV4mrw/s1600/logIllustration11.PNG" /></span></div>
<b>Evaluate using a calculator rounding off to the nearest thousandth</b>.<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiHZ1kbo2_N3FcvN39axXrcAl9uuB91jXhY4lxzbgo9OZoKqGq-oj9Cf4JQl9x8Y_KypHApRIwus-NmyZuTssMfAUe8GuJHQ8MSV8ZHUJy23zVc4WlL9MnEUD8p7EefnizpiXvJT3gen2E/s1600/ex19.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjkFxdkYu4fS4-wcsVUVBFN3Kqi-XmEzo-VRHS2z8_GZmr0BIyms3Q-6bYHLcs1Bc2Q9RNLOUaRwsSPloCIOC6Jthcme1TVN0vyamC1TbPMYj8BDSYCluHPyGCCKjwOOkwqGZ3YfGbEXT4/s1600/ex20.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgFVLD_iz6kzXCLZh-5fCNyDwglztD2yxjp6lmBjKaaB8P0hUY7hdQcP6XXlvTd_M3Ah9YNvrHvW3WISXAsRHwR94JRFmgQSq1l0NLlKRqfvdYHPK9_A90RYX2V0QFrSiF3nPW1ZewkNyk/s1600/ex21.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_Zxq-4jmlEIOTfc_f3ib8fhn-GaF5aBIKCEj7Au95psxvOBjq4sBTjUnNuKZrRhbSL20d4RIdp4p_RMGy-fyDfb6F4yy4gaTjJLAKDxBQnouqlJjRjClnTc0ScVnCHfPKrZI5fj0srrA/s1600/ex22.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhiIfkCnSFdyaRLAYyBtWCPvX9pm251RCq-HQQplb08kLslpI4_egI22c9oE3Mvfk5HJwgEVvikjOlyAWK1mY0UNew9xsdw8c-wJqLJJE-DyNJULhcwXIWqi8fiqMDaygg93fPGD0bXt4g/s1600/ex23.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiXtmlv9AspTHBCthMMp2WHBINU5rRJRnBu5yntNw-NhjMOcNYv1W2H4VXOm5i8rFR1nftm36aJEV6HzqOWfaXthmotW3Sw-HnxTIQ6HVF15ze9SVZvHl5LUOMZJfyHX3N-L0CnmFDVurI/s1600/ex24.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgdPUJCw0aHA5g_TVYpqPFqaj-wSGhBnYJx09Dksu7ZALHP-MXWZGKxS-A9mPspG3jdSZkJM30BEY_t71HGK80p1jiKOKX59AZK5CmHVmFQwqOgqfMmcuK_Tvo9cc7ctq8zol-5i38gzsY/s1600/ex25.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhgEfGr5wInGxsxl6lbaLFS0Hk3EXSzTtHDqaunWjJIl76oMI3m1KUG3O-mVuEHSz6nXKIqv78U5Hsh6-a5doE-LSPfq82Un2cdBW62xYfI9MSX2gbGDCpJU2zo05ZLFHjokxeBEejKNpg/s1600/ex26.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhBWYu0MgdyflaY3xggBdHKOWZ6mxTBlNOh9A8rhG7DSWcF3fUImkPr5r7UUW-X8vflqHdFWmppSHAr3d8zrYPrwrDu3sJDksyq8cfRhg2P23Mabf-_v3f1BLxPyzOfxj28fN89NkLk_Og/s1600/ex27.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg-pCidpw48Y1h2l72TLpTTZPZuCPVmucu0hNcm7xXnD-Ox0ErsWWj44sVaxNJvwxfGHUKNSUgqelATQ_uC0lEkQ3zauz0C1AAc1cpdpLMHesV__AQKuV45VR1ufmRYqRLYY3T1hn1FUDc/s1600/ex28.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhg8IWHCtJd0087zVLBhPfFPNsHIy8SxMwpQTG63RFs7irAy89D4vwDdKyZfywxgHk4KOepWNqUoDLBXTOddG4NEZwYS4wn2G4YoVXXT2ZC-rHMjd4ARrwgAULtdwSCd1bxqI64kMi1REo/s1600/ex29.PNG" /></div>
<b><span style="color: red;">YouTube Videos</span></b>:<br />
<a href="http://youtu.be/vYxiiimh9mc" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6MVwVIu6vLiSfa51LaoMmhHfpWv2UWPGeLnAXdlTKzo14if-n9wX7469sbNk5n5BqAdjdueZRAZnwLkJpQRZoJc33nS5F3ngHYhTXTvOiMpB540SL_KC4fQY2GaZ1E_gR7s8_3xncY6Y/s1600/vid01.png" /></a><br />
<br />
<a href="https://youtu.be/Uyd3mBofcX0" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj23WHurHD20jRbNvaVrt4WwpDrm-nECUzdGLSONKttTkvFY_Vp7DdAraAuiVUkBB296AiozcXOJ9LWP6U63bKgS_DBypo5TVngDoLVs7U45xIS07d7rvsMGwzYEX6v_f0g5vppkRRF3U8/s1600/vid02.png" /></a><br />
<br />
<a href="https://youtu.be/4p2q_nJJ-nk" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgzay708YE8S7siSzE5b2MvWOGgTAr2KrHqsEOUSrpnLp2eqVvWGc_Yre-BXsgcAkljbYDkDIEX_qvjF8Q_1pbz2app9aqNqd4YWaqDjp8_1tdYoe9vbkyBmGLFLZG1nocTAPx3Tn_nFWw/s1600/vid03.png" /></a><br />
<br />
<a href="https://youtu.be/2qEtpKOjXfM" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhWbY6qUq2BtluMVUliU9j7uw7yGP6hubmNxLEIFzGwONOjR2vigl7oka0fvH3eov5rQ8urrkfjMSetsmMSWWYoLhMsQ7a0_PVEta17cWtIZnFMwvyFfVPBX8ONLKaSncNcX3h5ZyLz-P4/s1600/vid04.png" /></a><br />
<br />
<a href="https://youtu.be/QR-gleX9rr0" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEibo5W2HeZzYo9saWF-CMYCworUKhYKkDfZzZvW2XeCS5yScrUnvi7rbdppa6NtbILJgiarwPaf0Wz3BoDzGYAnzYsn7IV792t79PkHoAnZS5_69bzlGky6bA_4iYEuSapGsDOtneh1EUo/s1600/vid05.png" /></a><br />
<br />
<a href="https://youtu.be/Wea7sq6LoRM" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiovW6iSJFuaxICLLwgTOPQZH-dX7o6_-rkhq1OmJ548vtUMw0x13FhhQKmr2253MoJyzkM-xzxQaEsS2ZxQjp1xyQuc43azJ0-XytPJJgyJphqjPBhnmKva9DG7SEL0MbmkBeQCg1Pii4/s1600/vid06.png" /></a>
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base and define and graph the natural exponential function.<br />
<div style="text-align: center;">
<span style="font-size: large;">YouTube playlist for steps covering <a href="https://www.youtube.com/playlist?list=PLmN1jmOiJEf5QTBICklHLjd1K6h4FRNrT" target="_blank">exponential functions and their graphs</a>: </span></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh8d9LkDrDvtLdSffcglQcdiE_k-5NS7198K9LspwVZxtoIn2Xg3jG9TpmGxTqgZIxvkpAu1X4IWPvNwJ1PuCIMEY83GSuWfFyWD8WeEVx1rVSlGTff8m1frkTj3r4z8CZlxOsrB7IPImQ/s1600/natIllustration01.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjG2Vvf6zSa93ONt1C6l7pIkVhlzhwaCcKDGUDpKE6FdpCkz3UQuQSHCYhnZj3P3fW9CCNSuFB4X_2DyK35oO79Ey1CPyHQWgH-9puwy7XnNZkikZPkFxr6obC8tygq4AS4XhSdDmocfX8/s1600/natIllustration02.PNG" /></div>
This number is so special, just as is pi, that it has its own button on your scientific calculator. Try finding the <i>e</i>^<i>x</i> button on your calculator and using it to calculate <i>e</i>.<br />
<br />
<b>Evaluate</b>.<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiI0THD-aZ9Sjkc40oFwx2T4RKQ8UMtOA0S23EZTUPBg3UQpLlR5bqa4cj5e9Fgmfir86sMXs6wBf781pL9wWIRmh0RRKX4Rm3cWG-AG1Uz1LJ2XpXQ6KnFSqfUM5B5dnjuk2dUIyIVT0o/s1600/natEx01.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjydSYggTngoNGJe11spKLDKgCTjFgylEHRJcpEKi9uhWCC9VdcZGkYubV6C-Iac5B9O_6UxN-dKODNv2i-5KHK33ATF_7Wz4QczQpk3BAHOm4879S8K8N4nefH1yQjyz1HdrOwiRHv-no/s1600/natEx02.PNG" /><br />
<iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/XGiz8YTEF6U" width="560"></iframe><br />
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhc1X7qi-j02hZNAO7zmqJ08HaaP82xjZlH0ETuniGChj-Md_z4tLT4ZTjlxYXkULgtVzCM77lw8nB_TveqeGu1CHuTtWqNM1gpPOfr9af_zzRyiet-WfosXPbWnK7Wtpf9ilGg4wwypFU/s1600/natEx03.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhzL1YJ1l1ZWRk-9HCqugic8jNkVJX7hYEKA5adBtf9ryKLR_Di3SHabFj9_Hc9aS8M_TfhY5VjgNZXoR_5LRIQcGBpKWxMIFnncAcwdRay8ouVn5ZW7rT_N0lRlvrT5EndB8mT362P7OQ/s1600/natEx04.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjuV4dd-f-aiJgjdT8FyPihNQ-QTSGPEnd5bnzSLBi53kghJowhgdfogTuHCozbWDGD_49INquqewxRudzWOpkV0zxqgFQcCauHa8sRt8_cTVYFNWSayWwjCBB75yYYxea3HYQ8fsghyphenhyphenro/s1600/natEx05.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgU9kovyl6bW_1IXIV4MVv6OdY-_lgLEInl1uhyphenhyphenPs18dpip_pXiINxhkHFuzNtB_GjpZu4EmCZo4kFVY-MFFLlO94advYV9hnzndvA6Bks0LwK62LYFaskp3LiKvbVvks66Wzh8L1-GByk/s1600/natEx06.PNG" /></div>
Graph the following exponential functions. State the domain, range, and label the horizontal asymptote.<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiUqqxMNMyRoKuGUHjP89jDX4UeS_s3BrgOlKz6lHDSdhIVOcngdnuStX68oeGjbTXL6vw6k9RsTVMHey0Ap0oSLkjqrpCWKETBOkGr8cgWz9GwpOsBhmfPY35IT_-ivlUy4CUrLOlEfyg/s1600/natGraphEx01.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi531Pu-uYC4rkdfvOEq3gtRDFJFJSiRvQWbnf1fuq102rUZEw1izqsUzvObIp-V9AVywJlPLThK71V7gOd35quF3T1EI3zQFuk3av-m_nXaNv_NQANRr0wu0JIyFFri_j5NqgpF3FyGOo/s1600/natGraphEx01b.PNG" /> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi-mLcboojAulc5PT0RUXdnN6sBU5WlHddsgvj-TYbhgM39VC9Q6uY8apcvES_2-JpF3pKWSXBlZsUf0QVE9joVO3w6qAdnhWvqJa12UQFP4A6B3NzX35dqhTNJS-qfa7IcM5cqyfNkI2Y/s1600/natGraphEx01c.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZX9ORZhlt-iGmiS9kEI7Gq8Ud0wjc08Xtp13ZWY_EvbL3gjCAL11IsXqNyaYVSMW_T9kWR1uD-5Ex45MiO3AxdJHlZbTMDMtKZ4uZb3TkWVnsTf8nPqqbr1b3h_miS2QUGlOlyu0ywY8/s1600/natGraphEx02a.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhIYQzvIg1CVphdqPiGMKiL4tonKIZBliqQ5geehk210XCxR-D5zKwS2UvDDBJOjAFJANR23uXoXhndFer37tNzW9w5DF78Got9HgMlNgbeKpQ1THq7GrThiacJNn9GugU8RB1K9hMDsjk/s1600/natGraphEx02b.PNG" /> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiLemPEaYkNswPvRAUE9CdG57kinnfvEBxo25lGiYzu033ave0F-6THCuupsFPq2W7qW6ga8y8hkbmTcBhX9p-OgVk26uZNLFqCo7I7dL3GtKioozzghJFIVvsyMvVP-mS9EZgaTZPLANM/s1600/natGraphEx02c.PNG" /><br />
<iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/zS44sz1KnkA" width="560"></iframe></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhNxMZLSBmJ8tJJF3sxFbudbvWY5-K8EJQUqlbUZz751Qn69q93qXyTnu4NZvr7a2HN4Yow05eEhInPH7tkDlRWTTrzWa_TLGRlN0pSWDzv1ZQ2Mrhgd-Obcv0tKHwq4Ww3WTcyK7BHfUY/s1600/natGraphEx03a.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiph-D9rZGvuRgr-prj5QyJHU9LrL-VS3ALv-gfrMpGTGPxBc5hFXeacaHwpytKqqK0h25B0XnfZBFWelWJfheZRMkQrsUK6tyQCRily2e3IlMpAGK5XCD_Co1UM1i6u5bpny51lhzzySA/s1600/natGraphEx03b.PNG" /> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhC320WQnskbrbzAVYpyRLLacTfhRW1kUuPfJeAZmGUJ9m0IoAYHVvFqDyN0kd_kgsJKhnga09ECmXnJWLi8hnBcMKFrro-9yVsx_58Hqy4GU8GFyfQLyxS9iNqFwDD2fETMUzK0kUm7FQ/s1600/natGraphEx03c.PNG" /></div>
<b><br /></b>
<span style="color: red;"><b>YouTube Videos</b>:</span><br />
<a href="https://youtu.be/BKS6ptVoEV8" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjx1rnWaOEd-iZVQxhSB2YLaumWh5FdEMnC-VyFQtZ22Lz5p7sbMJbWM1A3FBsEf1tPxjWSP-XUc6DuFRQln4w6zlv1-6ltue_nFAKgxxxACwyC8jKrzXIiuqIRjPr6Wks6P9y-Vfdhfcc/s1600/vid05.png" /></a><br />
<br />
<a href="https://youtu.be/XGiz8YTEF6U" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiyyLZrZZlo2uxfaiBAMprrzw6cDS05qBBhUVWNC1ESup_VDEMnrh9gWykAW0HTn_92FVMLne7hfQknpWaL9VsZipFcsMcp-Ztu07WeTF9u_STiEqVqmTImJM2mcx-8bBQaRemUgYQMX4M/s1600/vid06.png" /></a><br />
<br />
<a href="https://youtu.be/zS44sz1KnkA" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjAAydnoCI8RLwEs-d2kLbUDtYR4kzLOOwtH1zwMx0dveB_DEBYV5aSCNCjoH0w6sBm0VQxJQOOsh5eP7DdG_RvNHB-RLWmEm_3hACvnz6mHVs8M3tX7NyOMNyEv6BXqs43tMe7Eh5x7ew/s1600/vid07.png" /></a><br />
<br />
<a href="https://youtu.be/0XeFk8WJ6yI" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEguUbQpmuw1beuXun0OjtHzu6JsJ1mlwWhvmpEXsxdI3buqPhLT2BhAs-YWldk_WI3_kWgtwM4E0JDRB55dBvi5zu3gWTpz34bYhHozW-OuvR2ddMhkMseV1gDTpAhRpM5XkPewLgp8GdU/s1600/vid08.png" /></a>
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Reddenhttp://www.blogger.com/profile/02231088988057526726noreply@blogger.comtag:blogger.com,1999:blog-221233850938098964.post-1742261479905856222018-11-16T17:30:00.000-08:002019-09-24T08:11:57.974-07:00Exponential Functions and Their Graphs<div style="text-align: center;">
Up to this point we have limited our study to constant exponents. Now we will explore functions with variable exponents.</div>
<br />
<div style="text-align: center;">
<a href="https://www.youtube.com/playlist?list=PLmN1jmOiJEf5QTBICklHLjd1K6h4FRNrT" target="_blank"><span style="font-size: large;">Playlist on Exponential Functions and their Graphs</span></a></div>
<br />
<div class="separator" style="clear: both; text-align: center;">
<span style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjH7_9Xvy3_jaolLpUZHSiE4xnJnTNQT_AHWRajY0L6ndcWoK4VLyPRYtS3M2bAdyDlqm9HtGAxm46v8kAR3UlKRxP5NxBDaiw8RII_zM8_-GW6gVpwZNbyZ_2ZXOlumBbZM7dhj00Hu0Go/s1600/def01.PNG" /></span></div>
Here are some examples:<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhruNvndxNeNbQ7wZyOlY0J_rAY0KgIZYed4E8ey4dY7nqnpJ-RCJjV1hLceg1IxyPs23VN0YNsB-HZkCaqkUmta_zC79SbIIua4yfGP4qe_YXoGMLihSKCMLYqaeL0qTaSbC2S06aPrbIW/s1600/ex01.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjaO45t2MKZpX_h2uEBVwZgKk3PkazOy2lBmYlfTUn_SzxIW1mAgzi2PTOhQsNLN1s00lwp-Mkc0Lo7bdtkdX7O-LUhSsN9pLyBJRx4O4UWwoAdH4g29kAFQRBIMjt1EaEdRQYsRSsCNlTg/s1600/ex02.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh8CdtSnjlnAvurh1YT9ZfrKlMxDAT8ZB2ikfV1YYrSNItaLB9633rexybreJRpu54UYjn06sb20VXKFFEyTSnGyPuymxbLJ-BH4IYikS98ygR4IMKdHhKaqFRaiZSqw1rHn55Yo3ERBdu1/s1600/ex03.PNG" /></div>
<b>Evaluate</b>.<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhiWkQ28O8hqJ9T1jVl8RK4bhJ9vKOv6xtJvd5QVl-3faBl-LbLJePVDt0AbUnB0sRgWaHTChO1dBkw_04zdIN2F6vQkhyphenhyphenh-PJ_jL2n-B9ghaog4WwlPap35-yBI1LGohufXI0m5O4xiAh_/s1600/ex1.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg-Hr00oVagpaYfs9aezrT0eh5UtBPm3TqQtylrltPBfKwSi_f_xRkdD7KY2rjOMeTiGf8a-fAGGDkTACCpdUi_dqZObDFSgWEhyphenhyphenVqtTNo488eCBI1DvmF_JfiWcfMMuv4OJ6k1xrVAkBQu/s1600/ex2.PNG" /><br />
<iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/OVOXPzT2_M4" width="560"></iframe><br />
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj9uaYso4VmPjNIMjPbH29PuGZJ7u0don5puqRDammmBCrOJfwiFFKp3mz0Fz76PsekyNExHl0FAOsGDhASLBWZ7PNsJzaNSJvNWzI-nUAJ5hjVp0lTJn1G-2UTaJeyoO9gpbgVpXHYMuaT/s1600/ex3.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhdK_X_XvkA3F4bo0O13gq9F2_19B2sp7QOqnSJXVDn1zpKgAOcDcllzRbmCnIQW2rRAkamRaGk7lfsQmDtnv7CpGgzEB3BzLRHUYL-Y0x6xp1GwWtzQobnHp-roOqjVW4LXs8Wwn-vdkmq/s1600/ex4.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjSz8JVkLFzx7bObiywG-8LnitucajnzvGUPygl4-aU1n6cJVVPfeHTxBuuXELGDjQIsdjKgB4kbFw2pCJbXBXRS2T0aANzVSVqCNUYnZHtN09ZJ2CTCKPmr3YC8DUu2cEXiyH3nZQVvv8r/s1600/ex5.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh9RdF5_CFzyJPOKr0BRvVsYp-gvKXlgnngOXfOAZCDGTfPKrF0_Z9oyLpPS_y8NmvcqH6kkrHHwTmPldSJHd-tPkvwZNzj2sNeVnYuw42Z4Yxg4ZB8GlGO0ku5EvVZDHj8WMQyWL4NgfZ3/s1600/ex6.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjXdhwRT_KSlPhYR2u7_8TZUeCDPEe1X236jFr8zJW8pBU_ZKXkVbhwjF6HZv03R_Bn-4qbvENHv_tdzrxjf4MqER82cgonMQkfQmjx2qfRiwiyp_r5PpBa3BAc7et8TNWzAcG1yaBzM_2I/s1600/ex7.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEglrJEYPSKJXQNs1EqkGfnI0wZImTss9N8CNIClULIZi9mR0mPw7yDqX_p6WODFYWfkK2zS4Hi8V_e10Bh09E6pi15UNqGVZhJ7XEyytu3YGs8K8cMdrjxwyhPEK0MN-BsnQH9Pk-f2nXwY/s1600/ex8.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgusbMFp8kud_WJ8Obh1vtUz9JveOB6lyIj7WwNElWSQXaZ5-R7oWJ2_x9M024JZcXMbJteIz3PHDHWAnxqX3wDPoKYbs_AEmyJL5ltYfCROiuC4GAbfRpUwREgbF8cE2c4cnsd1tn_P4Fy/s1600/ex9.PNG" /></div>
When asked to graph these types of functions we plot points.<br />
<br />
<b>Step 1</b>: Choose any values for <i>x</i>. Choose some negative values, zero, and some positive values.<br />
<b>Step 2</b>: Evaluate to find the corresponding <i>y</i>-values.<br />
<b>Step 3</b>: Plot the points. The points you plot the better the graph will look.<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj7O3Gp5HEt3TJhNY-Oxn88ofJM63YHWd1d10IXSUrLvEihW4gQybvm7OTeSKxkjJm6anA42Kl1hAiOzRF9FylkBCXGQZ77rS-_kG8Vrs3-rBviIHFTRXvaSIpU2qy-I8l2bPaDbdGZQ6JG/s1600/example01a.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjvuF1sW_LSUM0F3j_FiZuYa53tYVm9y-iHDutxr3a7m4cdqibsczFJ7spyW46i5vhFYXwwJ-UM7z5YMyUebvGRXX8vBwNWXVvtAlmYKBUp3UMrZ2O8PfMMXy12z7wXFFk3MwXVzFWotBh9/s1600/example01b.PNG" /> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg5qRYhjTFo7rPfVdcZ8ZPdc1ILaQ5asU0-kp57NrAeCq3dBWUeGuVNweLiFmjMQz3eNjELKsTQiBLBXDr1Jn5eEHCnLg-a0C6FGuYr3YEkvfzRMb_UXr1NfDzGtDCawDA2mXYd-XmtZ3SD/s1600/example01c.PNG" /></div>
<div style="text-align: left;">
Notice that the <i>y</i>-values can never actually attain zero. They get infinitely close when <i>x</i> is negative so the graph will be asymptotic to the <i>x</i>-axis. In other words, the line <i>y</i> = 0 is a horizontal asymptote.<br />
<br />
<div style="text-align: center;">
<b>Instructional Video</b>: Graph Exponential Functions</div>
<div style="text-align: center;">
<iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/Sc6kQ3Eae0A" width="560"></iframe></div>
</div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEitdQ2WfAnhRl3-GZbfDFDZKToprz871Wk1LpR8j_u1wtl5oVqC3S6ORvoMbTtawMDCz6kddMqtPyY7jSZbpoo-jsPRfZygwEn1XZfgTK5koxPyLTx97tGcw0SlpOpXiCsr7KRWzysoU617/s1600/example02a.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjVxldlffkNoB2P_ULgAqubJXAQO5_Mv1K1s8xVrTnLbx8zuJkshKBUdtO1QVUE_vWB1Bn99zvRVcfXLwjZogD3YuOOBWkYwlbwggOPui90CTQULKxafXnkUXjHpBON94KgCUeCTV1wKnaz/s1600/example02b.PNG" /> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjr9-bkEqvMOYhPuOV2ak90KuLVWTYD7M0Xmz-Y0mKi__TloM2M6BCK0rWqSEytCSs9EmjfMWH6W_a09IpBe1MeCLn_auH8jGe8bN6CleFsF4cn3DqJJs9Vbic6cVxTe0_x-ADRfpnorzko/s1600/example02c.PNG" /></div>
Looking at the two previously graphed exponential functions on the same set of axes we will be able compare their rates of growth.<br />
<div class="separator" style="clear: both; text-align: center;">
<span style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi7fx4Ue-tNbXV7HvxFILCncbnwTS_rKVpnPBm7mNjWGFieVgezyrRda0UzhjdUSpBb3jPtzQZShKtNVjgGOibJX4x-2oOvl92i_1Ew2LePS9KFCZg9h9doUE7q-eTbY_FAkt5Une3gtGxc/s1600/example03.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRZs2CKhosFONjK-ViiAn_fOQBNNoaG_lMOAGjmu6ziHmZKLwYTP3-jO9JWm0cRaLLe_iPs582qoJU-iVVOy_wDOMt2KhzWM7-SEdlFRylSUvOoNwNKId_fVFIMv6-wKDLUK4Z_xknc0XM/s1600/illustration01.PNG" /></span></div>
Notice that for positive values of <i>x</i>,<i> g</i>(<i>x</i>) increases more rapidly than <i>f</i>(<i>x</i>). Also the domain and range for the two functions are the same. <span style="text-align: center;">In addition, they are both asymptotic to the </span><i style="text-align: center;">x</i><span style="text-align: center;">-axis (or </span><i style="text-align: center;">y</i><span style="text-align: center;"> = 0) and they both have the same </span><i style="text-align: center;">y</i><span style="text-align: center;">-intercept (0, 1). Now we will explore exponential functions with bases greater than zero but less than 1.</span><br />
<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhiH7BAIzEvArqt4WpENzXPugdtBRs8_fRGvPqmYv8QBx4j9fbKLuokpvLESCmXpN_GWsUAR4og9fmK_APMFhm5WQZJ97-K5fbPkeLXs3oIWBT_nYOQ6NAbH0M4hCfcOTANPaOFTk_c0ZPM/s1600/example04a.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhzPHRa-ClGmiRnNfAFY9pOROkt68f0eeKg9_2THi6GeL84hsToBVn3-dbXLRTHsL5sO97GtG5jtEc6MZSSYsOsXNOARufopA5wrNP_NBL6rjQv0vIXSCgY9sONa-ZzQ_eM2HeBBO13HEB5/s1600/example04b.PNG" /> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiERwRBqSEsmtbkZeeBti35t8o_f_fI0dIW3ILVu1DOVGPjNZbp-K-BE-YzH74V1PT2qqJ8uAv5_ykETqLOOabBLbQnyBWCvzCmljQ6pHhcSgcex-nzqj3-Fhz7JNId38DCz05mCfLOZY5N/s1600/example04c.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiXtKTv2eT895PuIBNUNnGzKYa8TO89ATgoYG9pOih5eoVeCiYl_LBexY50_j-1gHz67Yj3tJvHAeZpAOXbysYdBrFHtTJoFhvPJ1N_kAPS22DoN9cSPa6svyKggfxwY3kUFmJz-Pg-a_6P/s1600/example05a.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgJ4DVdZbzd4zbI3VSyCOx4GqOJ3IFtNgXpDbl6BBbdyhOnvvdVR-da7HFaVqoa8FTXMMEKC3D6aiabjEcErAAt20B2cuel4abo4bRhf15sDhSe0m9ZqvXaXGL_kTQ1biCKrefQJYcUdulz/s1600/example05b.PNG" /> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhe8uLezzPoLCmkz-v2yBz2hw6i2pcONF1PSZoa4SWSVN6RsNmj4vqDxMnuknF1VB-tsc9ij2ak4CqupGI-pBI-36_wRDPBSZsS-hF0Vnuhcu3CCUHptuJJr5Tqqz-X5nLRin6scfjt7DxY/s1600/example05c.PNG" /></div>
<br />
<b>Tip</b>: All exponential functions of the form <i>y</i> = <i>b</i>^<i>x </i>have (0, 1) as a <i>y</i>-intercept, no <i>x</i>-intercept, and the <i>x</i>-axis will be a horizontal asymptote.<br />
<br />
Looking at the two exponential two previous exponential functions on the same set of axes we will be able compare their rates of growth.<br />
<div class="separator" style="clear: both; text-align: center;">
<span style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgttVb1OfZDVhRi9aECiRgPZLNZIFpPCzjgpN7Wm3q2a4_R0QmypPzWagxx1gKiWoJ1mdXmGn8m_z2ANR0BTwTCHekt6arvTVelSdnlqWkaFgOT4gttZdHq1jVMKo6GrLzCL3kAjdPvVWz8/s1600/example06.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgRZs2CKhosFONjK-ViiAn_fOQBNNoaG_lMOAGjmu6ziHmZKLwYTP3-jO9JWm0cRaLLe_iPs582qoJU-iVVOy_wDOMt2KhzWM7-SEdlFRylSUvOoNwNKId_fVFIMv6-wKDLUK4Z_xknc0XM/s1600/illustration01.PNG" /></span></div>
Notice that <i>g</i>(<i>x</i>) decreases more rapidly than <i>f</i>(<i>x</i>). Also the domain, range, and <i>y</i>-intercept for the two functions are the same.<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhJg7qU7lJFtUARR94lq-i7aE9psNFIlRE4tKQHsiof9dBh12f_KCEdLc2dsErUFRSKCd4CBiZ084yZEoggXunZbBRAJimyaxg2X44A9Vf96tQsXL-StICZ9YJJIM_v_oNX_9IlvxJMxuAt/s1600/example07a.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgsBHfv0RahEqrOttmXHCBSJHdD4RPZAKj8nSNlMYOerxJhMBtu_Z1WpLG9d1A__Fz9HpGtCHDc0T2UOI_GW8794d5T3XHVoGBC8RKD2tRLTsUpkDiBHwrYOQln-TAPUBba-P0tWr1jFPMf/s1600/example07b.PNG" /> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEidn58HPxO07jyWMyU_RjWSDTN0nSXUiOmz_TUeY2Qi68XzAdziAi_L2CWI0wQAxX3aJEUC4O-DC3jH74oNL268CFx0p29F-HlSDyhF4lhhCUlvWBSS0zkmNn3FwWICbAM6vp9s4a8Ks_LX/s1600/example07c.PNG" /><br />
<iframe allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/ptGDoW4Rs0U" width="560"></iframe></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgj9437wVp7KM0sKuLY2YPHaLjDu5r4QztX_veykFBQD1r9Kvez42n3umRtR51vZ6pkce4nK9v2CQofpB6kWu9XOCGOpkK9MKldx9RWS8PIrL2IIX_yVzswA11hJxFhYDhsAtf60l0yjfy1/s1600/example08a.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgDuUXrq-D8qoPjH1c3bWa32l3aMJT17fEM0yiE9obUwgDoDTrwddNuJ9V0s6gGGGsbvv4ZOSpbDsL1UxzmAecW-ncCaGKuukpyr9ypQTVFIbrliEtZcIz0Gy4oFq6qtqBicBmuCoBzUZSE/s1600/example08b.PNG" /> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj5dJEj1AoH8U_SevCWWiHoEnhIr3KjX39FXCMRXRDeVUjTwECesTgLuWcpliCgYO883oY1PKSnj-w6cu_EprpSCtuLyKXdoUKcdILe3C5u6iNlVv8Zox95OR1nYnHtPaSApAQmVJqjFUdT/s1600/example08c.PNG" /></div>
<b>Tip:</b> All exponential functions of the form <i>y</i> = <i>b</i>^(<i>x</i>+<i>h</i>) + <i>k</i> have a horizontal asymptote at <i>y</i> = <i>k</i>.<br />
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgNE36S6ccvrZSEVQuyJ6STO7V773lv0xASYmN6648gLoRPR97OYzzJuyhTA0OlnaSk8gRJI42hFAb-pFDK3qCDZW2SYfkUJiXbw3jD68YSpToXoulY1k3l_Hdcnbrlhj-zTcgRN3DHjWNI/s1600/example09a.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhVQkjqhMVCV3fV2A7zGpLCeBJ_5HyF5NcV1f3F0iVmqZ7e9YgEEgTHElctI9usRaX7WIQU90fEEo84DzfEaC4r6CAOZ2_IZ5M-UiryXkG25K2gZobtzIfXnQ6xKWkyVQf7XKRdlSJ583A6/s1600/example09b.PNG" /><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgApmJsT2MdKRnvg3hURf_gXkH_xjtOfZ5d8vXR7CYumAl0k4WgS1ZCCUWYW4ao6HIRfGP_sFalLnlFp_nGblb_kWaAL6BcGS8XZ_VX63nWKthBgsSHNhshoQRl_hRPHkVKc_vYszax8iyq/s1600/example09c.PNG" /></div>
<div style="text-align: left;">
It is also useful to note the rigid transformations in the above graphs. In the previous example the basic graph of <i>y</i> = 3^<i>x</i> was shifted down two units and to the left 1 unit.</div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqvQIdOjKu_zf2MaVLTu-zQ_JTUrVoy1kerntmSdLp8p0mSEk57rlkfz2Cip_8ckk6lXveoNH6JVzNnMhPin4tKOwwShS_wR9NBtHbIkiaRgmpuIBUf7S9Ddda18oAuaVnfGuUfRYxdyFz/s1600/example10a.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgqMwJYIPP1V9vQSiRyJ0hmG0lSOwzPA6XE_XUqrzykyEVx2jRNp5L6JcnKTh8Iz8jZ3k07LPoC6PJLJ5xKq_3bKw4YqyGixtGWC_54SedZP5qUEqUbydMY3hQNpYySoSrPkXSmmQE0qkQr/s1600/example10b.PNG" /> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjzyBXzwusgFkGwAhzOQu99ff0KH2jgKemU-mgadySSqHU2MY_b5WepsUpOMtWAOdmuc6G0dpltUUESS-gv3ne8my-tHF5JyICSXfuKmVp6DZfwkxs7vKP2ZVQ5QqGT6fc78DZ9gWuQpDrJ/s1600/example10c.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiBIcrbr2RQQLlfrJoSarGRkUJLUt3Kv3sE5MRHmnIl9x2VoJ_WONBTdJqEro4nFvmMmET6nJXWiywp0OyucoOZVnIeaGWTHDn8F2aUmSjee1iqB-RZFwK90zQnhpKrh6RyQ7eZTyuJU4qP/s1600/example11a.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjHf7KmqENq1arHXC9z6vngEVhXnBXAJPtPEXl9wadcSGQTkb26ZmFiF_vGvKC0c4R84CX0TpGTu0Sn_WZq9lDc3b2cfBQkWkvsuGwFvMpFCd8talt9GmfLaJeEArZmMyz-Xx5mXIXTG0pG/s1600/example11b.PNG" /> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhgPz7y4Ygv4uw0iQ6CDYkbI_0C1CM6s_G7qK8c_XqdckDpKxZWoryzbtIYSKWuenV2f1Jh94ybg5uYdvArfMKgb8BFzVUXAFhh1-AavyHtsxDjN4JWAEteuYd6h1IAn84TC_qaL__LGczJ/s1600/example11c.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgnf7mNRKWyu9v_LatGVt55b6wXW5G4IeIlbwnalhUbbosqLXb6ujK4416J6VN_EdhLFrFTjEXDG7r_cTgCGzdB_CMj9LZi9MKjhyq0LLPCD-yeJ0rC8eipvAcivSv7Nnr3m06ByxgGLjAE/s1600/example12a.PNG" /></div>
<div style="text-align: center;">
<img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgOKNrQGL2VxdbtK0rOeBkl9CYnHsvDni1k5HZmKtGnG4G2EJ3ASedoqmOfB0HyVHq_j6G1vsGXEGiDuNg10kfU30jLsQxnaODn2A4Jzn2fIHj7wxXWDwQxeg35P8wG3hOcUfst39bhhjEA/s1600/example12b.PNG" /> <img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgtL5yTFt03Idt7yYXqWO5qFIfgQ2HbKz5-KrfB3qqmR8z09kgqlHYTlO-Ab0CWl_D8q5hGC9nGBaMapkfwRVjBNhADIbz-p8ysl9NxpFco7nthG6ZfwJPzJAaRbhdL2azZrxlCAyPBOhe-/s1600/example12c.PNG" /></div>
<div>
<br />
Notice that all of the exponential functions graphed above pass the horizontal line test. Therefore we may conclude that they are all one-to-one and have an inverse. This is one of the most important observations of this section. We will define the inverse of these functions in subsequent sections.<br />
<br />
A common error when stating the range is to include the <i>y</i>-value that defines the asymptote. In the previous problem one might think the range is [ 2, inf ) but this is incorrect. The <i>y</i>-values get infinitely close to 2 but never actually attain that value – this is what it means to be asymptotic. So the range is ( 2, inf ).<br />
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<span style="color: red;"><b>YouTube Videos</b>:</span><br />
<div style="text-align: left;">
<a href="https://youtu.be/bnkoYuMKwyg" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhexH2nHfwhJCepBdRr08qPlq3Gjja25Hp3-_R33gYmqlsETGSdSzF28UG9uFxqw4oETVgYwOR3BHvTZhIxASOBVXyipZ2hH1nBjXG-E7gW6i_CMDS5geQPPCjkWmnszq7M2xoAqgaw8sEW/s1600/vid01.png" /></a></div>
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<a href="https://youtu.be/OVOXPzT2_M4" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjwZQ253E7AphUJfqZThlu61iTeEvOGv3x_8y4OKofC65z_ODlwBpFng0V762662Aho72HgGmwQsZ_XNbg48jmwNY41Cjj1CyrhJmtUbgXq5ATIIGtBm3iLCe6AC7vcjws6qHZ8qvVAxi8W/s1600/vid02.png" /></a></div>
<div style="text-align: left;">
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<div style="text-align: left;">
<a href="https://youtu.be/Sc6kQ3Eae0A" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEinMSEKysTYtBLXw7MPZ_aX2tslgsGiRLHh_ratDvL83xjIsUfamA98xefh5LCJsX-v7h-IMMXN7jmOgaLJndCUImCB1aMiSqNHitNfGgs24rRyHA4iazA3Gghw5PU8bakrvaCsiIGOEVbk/s1600/vid03.png" /></a></div>
<div style="text-align: left;">
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<div style="text-align: left;">
<a href="https://youtu.be/ptGDoW4Rs0U" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjqrSZ9DFv8-TMvA6YdTIR3c7b2CecmIs1gXEVBFkbdLYfvhwM226nsrJz5cMiFppgxn7HEJmOErztUZTEs6QNikmSlYVw1qtiexJS67C-KVwO86Xo05ljLkwaqc0QKGfmCr5t_0eWTCzd9/s1600/vid04.png" /></a></div>
</div>
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Reddenhttp://www.blogger.com/profile/02231088988057526726noreply@blogger.comtag:blogger.com,1999:blog-221233850938098964.post-2872444026204848242018-11-16T08:26:00.000-08:002019-09-17T22:31:13.137-07:00Visual Real Numbers and Their OperationsChapter 1 video compendium:<br />
<br />
<div style="text-align: center;">
<div style="text-align: left;">
<span style="font-size: large;"> 1.1 <a href="https://www.openalgebra.com/2012/11/real-numbers-and-number-line.html">Real Numbers and The Number Line</a></span></div>
<div style="text-align: center;">
<a href="https://youtu.be/NNsdEV0jJqM" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 12.727272033691406px; line-height: 17.999998092651367px; text-decoration: initial;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh95r5CedHy447oQ8L5-gwPf-FTkv0FKgPEfweCN3A6uOKpfT0-6FaPBnGFCXtF-pvj2lKapiek0knZxCnn7Uo_1_j7XEOICYe7S7dMF4ggfcS7c0CvLdNQGZcAegORGm8xsFme_M_77DY/s1600/vid-0101.png" style="-webkit-box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; background-color: transparent; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; border: 1px solid transparent; box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><a href="https://youtu.be/4UaR4f4wNuE" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; line-height: 18px; text-decoration: initial;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhuv52LPsAmnfy2xCv3lX4EIKdxo_KzGsFerHmJ5kG7Y2tz5bBMDCddtE7vClt1-tG8xbNBRW6-L8ZTH8tN45CsnuBHAn-iW4RpmS7OOlmqOsMiXgsxV_EnPvPtGFbBF-Rd0s1W5RV8k5E/s1600/vid-0102.png" style="-webkit-box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; background-color: transparent; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; border: 1px solid transparent; box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a></div>
<div style="text-align: center;">
<hr width="90%" />
</div>
</div>
<div style="text-align: center;">
<div style="text-align: left;">
<span style="font-size: large;"> 1.2 <a href="https://www.openalgebra.com/2012/11/adding-and-subtracting-integers.html">Adding and Subtracting Integers</a></span></div>
<div style="text-align: center;">
<a href="https://youtu.be/l2y4rLS8I8s" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; line-height: 18px; text-decoration: initial;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg7xwgnazCRi7aZz93aG1yy24zYe80N4Cmbg4DPNyoGaymjys2zvK1Escot53eSK0WoOlRWp1R9LEC21-001oX8sUsV6ULNp8Mnf3l1Ne8kvNqLmhXDr31H5GTQOS6WbYYqvTs9Jg5zHAA/s1600/vid-0103.png" style="-webkit-box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; background-color: transparent; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; border: 1px solid transparent; box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><a href="https://youtu.be/fGCRnW_i6I8" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; line-height: 18px; text-decoration: initial;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhoXJ-wFPW4KAdTROJx9_lZMxQOjNtW8PjEWVOPU8VynE__4TwM-uVxvekf2Tq-E81plgXPEHsBWZ35_32G6JmfvDkGWnQCeopGg4nxp1IEr9X2HzmM6-3nwBF0gyhcEytF3pziDY-4_VE/s1600/vid-0104.png" style="-webkit-box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; background-color: transparent; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; border: 1px solid transparent; box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a></div>
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<span style="font-size: large;"> 1.3 <a href="https://www.openalgebra.com/2012/11/multiplying-and-dividing-integers.html">Multiplying and Dividing Integers</a></span></div>
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<a href="https://youtu.be/ftzGHykWq_A" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; line-height: 18px; text-decoration: initial;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhTKcJxZHfg2woOLx3SppgiDMegpU0-GuzH6DRjhPoOZQge9QOe5WT-1JQzWGxJcnI6gVqFF9IfQ4jCDZpPNSXlOyG0sstFX0kWL8ApoDpfFKejqbgEMwm9kORLNo1L3JtvpjiCrrq6oxU/s1600/vid-0105.png" style="-webkit-box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; background-color: transparent; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; border: 1px solid transparent; box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><a href="https://youtu.be/mk2HoiqanmA" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; line-height: 18px; text-decoration: initial;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrjipxlx2P1TopcKp_RQTSnm5Nf0l0iftWkMESO3gmoGkFbC8URw-6IBfLMiSpFmY3ORbPzVq36lIaSMiMRvqDtptfvZr1UxEBW9scGvQAAhz2wU6htjt9t_JvzPhGfTrbok3MSy9gCBA/s1600/vid-0106.png" style="-webkit-box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; background-color: transparent; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; border: 1px solid transparent; box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a></div>
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<span style="font-size: large;"> 1.4 <a href="https://www.openalgebra.com/2012/11/fractions.html">Fractions</a></span></div>
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<a href="https://youtu.be/D1JgMzcwC74" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; line-height: 18px; text-decoration: initial;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhgX9BDwVd4VzH2LofmVDYQ4THFW4ZPJ8klLlgxarQgxGFSeXqNJen7I0YG6TZ1soSN4Q0u8GLscmY1OkUim_hxVj07gelCQfFNffqdrlUzn0DoL3x7kFLdzV2BPXBxzQkO_CuGCaEvmPE/s1600/vid-0107.png" style="-webkit-box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; background-color: transparent; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; border: 1px solid transparent; box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><a href="https://youtu.be/0nLzm9AWBDA" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; line-height: 18px; text-decoration: initial;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgWvvcovrebKWU1XxHWNEUC6JGJeOxI-ESEm6xjl6cu49mzOnwMSvU_qH1kVpldUWM_vrG4khlMK0iaYfQsh5DvG-xVSAw6uTWapztFHCjpMmx-32cKpJsJ7lMXsrYCPyMk1JUnHfT0bqI/s1600/vid-0108.png" style="-webkit-box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; background-color: transparent; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; border: 1px solid transparent; box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><a href="https://youtu.be/T47YNYsxfp4" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; line-height: 18px; text-decoration: initial;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg1e2m4z-wIkKxVzpOn-mz5LrQpFIcfzdibbEE-R_wKjvMqgQIB6HP3KAvA-8yAT0QdO_pAbAkWchdtLyAuv7alZvw7njx-_GgixpPnBJqClG2ZWV5lcdC0bY4np2VXNS-c1iL7RWKB8p0/s1600/vid-0109.png" style="-webkit-box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; background-color: transparent; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; border: 1px solid transparent; box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><a href="https://youtu.be/ojuJs-2YAMw" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; line-height: 18px; text-decoration: initial;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiHCb9FCN40GYwoAFHpxqYP6JdkARz7UrhP8PeLMyspq7hCXWTgMhaup8J3CyCkDHo_cinEp_Tp-LR1h37xB3iIgiOmGwnw4Ima30MM3bM94E06ZUxCkQdoWGBFTDjdFjUPMbLK71WnzO8/s1600/vid-0110.png" style="-webkit-box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; background-color: transparent; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; border: 1px solid transparent; box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a></div>
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<span style="font-size: large;"> 1.5 <a href="https://www.openalgebra.com/2012/11/review-of-decimals-and-percents.html">Review of Decimals and Percents</a></span></div>
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<a href="https://youtu.be/NG8DrPm5qOc" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; line-height: 18px; text-decoration: initial;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi6xafV7303_Kba1TVk92mFOwyju3qVnrETUWFVUmmjDN8UTfVLoSrrIafoXuLE5nJYsaNKicnncPt86rbwh96a9VaSvpXb4yFkJsm3bqpyc0kA6NReKBzvSA2tUrfsJTjqZIsA3MxbUp8/s1600/vid-0111.png" style="-webkit-box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; background-color: transparent; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; border: 1px solid transparent; box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><a href="https://youtu.be/8Tbwx7ZdY1U" style="background-color: white; color: #3778cd; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; line-height: 18px;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjWL4Thahu1wX0gtyT4ByTkfNBZrMCNU8Kz0R6cijx_XKrPwX1NOuvesWro1VboYJugaztoDCfupZR6MDD5qmb4pQ-226iTg3bGCIX87heS0LLpkiAbiwQXKQ8hHwnQa9qs9bD60VL9SDQ/s1600/vid-0112.png" style="-webkit-box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; background-color: transparent; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; border: 1px solid transparent; box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a></div>
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<span style="font-size: large;"> 1.6 <a href="https://www.openalgebra.com/2012/11/exponents-and-square-roots.html">Exponents and Square Roots</a></span></div>
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<a href="https://youtu.be/VhJyEUdBpkk" style="background-color: white; color: #3778cd; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; line-height: 18px;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEguEcSxyVh4Lk2DJZgFTQAbhtntGL4J5yGddpSg8fLtmkdVS2UOBz_OR2yelQXltgvn-W8nFVbE3hYVeAKLjp8hDLwEeSCweiz6iMbsB08UlBdNNtw4C6RQ1KbtKtNjPc7lb4WoaDGcC9M/s1600/vid-0113.png" style="-webkit-box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; background-color: transparent; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; border: 1px solid transparent; box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><a href="https://youtu.be/6_QE-xrGBXo" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; line-height: 18px; text-decoration: initial;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjOZtbGiq1pe_7HrU5DnJeBXqR_wn2AlL1e5f0XWuLTlMkQVHNi3IzXa-4n_XmBJXIGSGGYQn-SXWGl6JaviDkm_0ns-kD9tMsQ_J3vcs8pYLaWvNUsBhgLtYoF5p1aojX_N0RLX40m_Ko/s1600/vid-0114.png" style="-webkit-box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; background-color: transparent; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; border: 1px solid transparent; box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a></div>
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<span style="font-size: large;"> 1.7 <a href="https://www.openalgebra.com/2012/11/order-of-operations.html">Order of Operations</a></span></div>
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<a href="https://youtu.be/3pT8gc99M0U" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; line-height: 18px; text-decoration: initial;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgbMFgKMi_OW5ZpjLACc8hmnaLTye9nwGcmPjgtuRY2G3nTW41FFjn7GkXVWXXCeJmX2pNPjfzACpM_71txYR8x-w8aa-E3iBeBO4Q1cTmmpw3S5EBRsn3B8c5rHt9RxV0OZnXTQcIJAZI/s1600/vid-0115.png" style="-webkit-box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; background-color: transparent; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; border: 1px solid transparent; box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><a href="https://youtu.be/D4s8YPGQBCI" style="background-color: white; color: #3778cd; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; line-height: 18px;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiDAohQefENZw8kc75zLoYj7VLyUoMXmJjRK_73_6ILDiUbJmo_2GvSbXri8-4ZtL_wQow3Lt4fFdqyVpBZQg4eJFXliUmhp9tABjQCknePBN3wH15jozBxpCJF_Obl5WUMY7dAn-JMB4o/s1600/vid-0116.png" style="-webkit-box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; background-color: transparent; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; border: 1px solid transparent; box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><a href="https://youtu.be/LDwmFzFx8ts" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; line-height: 18px; text-decoration: initial;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi_3CGZS5-TkoPV7RRMA7h_tKeqGeiDn6rq6_6wHSG0L49BynXPEnI_1mIr4wGat_-uxesrgtn2Bz5rxXHWS9f3sIUxxhZYEgDPDbm9Lu3JxkxBzkCIYXknxX1daWp7BaS8P_TmYVJ9g0w/s1600/vid-0117.png" style="-webkit-box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; background-color: transparent; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; border: 1px solid transparent; box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><a href="https://youtu.be/YGm4QPqDNSc" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; line-height: 18px; text-decoration: initial;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhoCTFoizBGessjROqwgPifSEU-enoHarjJ0dP9dfHa4WMaSMQlj7qPYmU3KQk2Fe3Bf4SVdpDOytRlj5Z_cebjAnaEQauByWYsrNVUHm-iN1JHpB1Fa3ONlCF5X7If5VSES-LXWD44m0w/s1600/vid-0118.png" style="-webkit-box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; background-color: transparent; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; border: 1px solid transparent; box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a><a href="https://youtu.be/JjvMPHaN7lI" style="background-color: white; color: #4d469c; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13px; line-height: 18px; text-decoration: initial;" target="_blank"><img border="0" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjRhao3Q2HSniWKSFCTkChiXZeosAP61K72jl-PLXHJmEg4lxd4AGjZlqHDrIiSbtlQCglRxnzBpVOtp-r6P0G0edlodaWyQOcndRIXDgEy8T0QRRzDr4_ZFHKvZOK2bwjlYfOEq2ImUxU/s1600/vid-0119.png" style="-webkit-box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; background-color: transparent; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; border: 1px solid transparent; box-shadow: rgba(0, 0, 0, 0.2) 0px 0px 0px; padding: 8px; position: relative;" /></a></div>
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