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

Multiplying and Dividing Radical Expressions

As long as the indices are the same, we can multiply the radicands together using the following property.
Since multiplication is commutative, you can multiply the coefficients and the radicands together and then simplify.

Multiply.

Instructional Video: Multiplying Radicals

Take care to be sure that the indices are the same before multiplying.  We will assume that all variables are positive.

Simplify.



Divide radicals using the following property.
Divide. (Assume all variables are positive.)
Rationalizing the Denominator
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.

Rationalize the denominator: 
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.
Notice that all the factors in the radicand of the denominator have powers that match the index.  This was the desired result, now simplify.
Rationalize the denominator.
This technique does not work when dividing by a binomial containing a radical.  A new technique is introduced to deal with this situation.

Rationalize the denominator: 
Multiply numerator and denominator by the conjugate of the denominator.
And then simplify. The goal is to eliminate all radicals from the denominator.


Instructional Video: Dividing Radicals

Rationalize the denominator.
Video Examples on YouTube:









Order of Operations

When several operations are to be applied within a calculation, we must follow a specific order to ensure a single correct result.

Order of Operations:
  1. Perform all calculations within the innermost Parentheses first.
  2. Evaluate Exponent expressions.
  3. Apply the Multiplication and Division operations from left to right.
  4. Lastly, work all Addition and Subtraction operations from left to right.
Caution:  Please do not dismiss the fact that multiplication and division should be worked from left to right.  Many standardized exams will test you on this fact.  The following example illustrates the problem.

Instructional Video: Order of Operations - The Basics

Simplify.
Order of operation problems get a bit more tedious when fractions are involved. Remember that when adding or subtracting fractions you need to first find the equivalent fractions with a common denominator. Multiplication does not require a common denominator.

Simplify.


We will see that some of the problems have different looking parentheses { [ ( ) ] }, treat them the same and just remember to perform the innermost parentheses first.  Some problems may involve an absolute value, in which case you will need to apply the innermost absolute value first as you would if it were a parentheses.
To avoid these unnecessary mistakes, work one operation at a time and for each step rewrite everything.  This may seem like a lot of work but it really helps avoid errors.

Simplify.

Video Examples on YouTube: