Showing posts with label divide. Show all posts
Showing posts with label divide. Show all posts

Tuesday, November 6, 2012

Complex Rational Expressions


It turns out that we have all the tools necessary to simplify complex algebraic fractions. The numerator and denominator of these rational expressions contain fractions and look very intimidating.  We will outline two methods for simplifying them.

Method 1: Obtain a common denominator for the numerator and denominator, multiply by the reciprocal of the denominator, then factor and cancel if possible.
Method 2: Multiply the numerator and denominator of the complex fraction by the LCD of all the simple fractions then factor and cancel if possible.

To illustrate what happened after we multiplied by the LCD we could add an extra step.

For the following solved problems, both methods are used. Choose whichever method feels most comfortable for you.

Simplify using method 1.                                 Simplify using method 2.
Video Examples on YouTube:



Multiplying and Dividing Rational Expressions


Multiplying rational expressions is similar to simplifying them.  We will factor then cancel.  Only cancel factors that are the same, or opposites, in the numerator and denominator.  Recall the property for multiplying fractions,
When multiplying fractions there is no need for a common denominator, just multiply the numerators and denominators and then simplify.

Multiply.
The previous questions did not ask for the restrictions but we certainly can list them anyway. Look at the factors in the denominator to see what values for x will evaluate it to zero. Remember that the function notation that implies multiplication.
For the given functions find (f *g)(x).
When dividing fractions there is no need for a common denominator.
However, the above property reminds us that when dividing by a fraction the result will be the same as multiplying by the reciprocal of that fraction. At this point, we will add a step when dividing; we need to reciprocate, factor then cancel.

Divide.
The list of restrictions in the previous problem is a bit more involved. As before, look at all the factors in the denominator, even if it was cancelled, to find the values that evaluate to zero.
Look at the denominators in each step to identify the restrictions.
Remember that the function notation that implies division.
For the given functions find (f / g)(x).
Video Examples on YouTube:


Sunday, November 4, 2012

Negative Exponents

The quotient rule for exponents can be used to define negative exponents. It might seem strange to think of negative exponents, but we need to know where they come from and how to work with them.
Factors in the numerator with a negative exponents move to the denominator.
Simplify.
If you are given a factor with a negative exponent in the denominator simply move it to the numerator. Use the following reasoning to justify this.
Simplify.

A common mistake is to multiply the base with the exponent when it is negative. For example,
Avoid this mistake . The correct solution is
Another useful property involves a rational expression raised to a negative exponent.
Simplify.
When simplifying expressions, it usually is best to simplify within the parentheses first and then apply the product and/or the quotient rule.

Simplify.
Scientific notation is an application of negative exponents. It is used to express very large or very small numbers.
An example of a power of ten might look like,
Use this to convert number expressed in scientific notation to a decimal.
Remember that we can obtain this same result by moving the decimal over six places to the right and filling in with the digit 0.

A power of ten might be negative,
Negative exponents appear when working with very small numbers.
Obtain the same result by moving the decimal over to the left 4 places.

Express the number in scientific notation.
Choose to place the decimal so that the first digit is between 1 and 10.

Multiplication is commutative, so when multiplying numbers in scientific notation multiply the decimal parts first. Next, multiply the powers of 10 using the product rule.

Multiply.
Divide.
Video Examples on YouTube: