Simplify Radicals

There are three rules that must be observed in order to fully simplify any radical expression:

1) all perfect powers of the radicand must be simplified according to the product rule (explained below),

2) the radicand must have no fractions, this can be easily circumvented using the quotient rule (explained below) and

3) no denominator can contain a radical.  The third checkpoint could pose a problem.
This “problem” will have to be resolved by a process called rationalizing the denominator (explained below.)

(1) The Product Rule for Radicals states that for two radicals which have the same index, you may multiply the two radicands and place the product under a radical with the same index as the two original radicals.  For example, .  Likewise, you may “factor” a radical into two radicals, each with the same index as the given radical.  For example, .  The use of this rule allows for simplification of radicals.  For example, .  When we simplify radicals using the product rule, our goal is to factor the largest perfect power that is contained in the radicand.  If we are simplifying square roots, we are trying to find two factors of the given radicand one of which is the largest perfect square number contained in the radicand and the other factor containing no perfect square number.  For example, .  It is important that you find the largest perfect square factor.  If you do not find the largest perfect square factor, the radical will not be fully simplified.
For example, it is true that  and that  is a perfect square factor of .
However upon observation of , we find that .  So  is not the “largest” perfect square factor of .  In fact  and  are both perfect square factors of 72 and therefore  is the largest perfect square factor of .  To fully simplify this radical, you must factor the largest perfect square factor; therefore, .  To simplify a radical containing variable/s, factor the greatest perfect power of the index which would be the greatest multiple of the index and then divide the exponent of the largest perfect factor by the index and remove the radical.  For example, , but .  When radicals containing variables are fully simplified, there will be no variable under the radical which will have an exponent greater than or equal to the index.  For example, while it is true that there are variables with exponents higher than the index still left under the radical.  To fully simplify, .

(2) The Quotient Rule for Radicals is similar to the product rule.  If you have two radicals under the same index, you may divide those radicands under a radical of the same index as that of the two individual radicals.  For example, .  Likewise, .
Simplification of the radicals would then continue as above.  These properties do not allow for the addition or subtraction of radicands under any circumstances.

(3) Rationalize the Denominator
If the denominator of an otherwise "simplified" radical expression should contain an irrational number (a radical that cannot be simplified), then it is necessary that the denominator be changed to a rational number.  This modification is called "rationalizing the denominator."
For example, if we were to simplify the following problem with the following result, then the radical expression would not be "fully simplified" because there is a radical left in the denominator.

Example 1:
                                  ... Notice that the result contains a radical in the denominator
 

To "rationalize" the denominator multiply the denominator by so that to maintain the equality of the fraction.  So now we would have

                                  ... Now the fraction is "fully simplified"

 

Example 2:

                                    ... Notice that the numerator is also a radical, so multiply
                                                                  under the radical to simplify the numerator
 

Example 3:

            ... Simplify the numerator and denominator
                                                                                     and then rationalize the denominator
 

General Algebra Tips

The views and opinions expressed in this page are strictly those of Mary Lou Baker.
The contents of this page have not been reviewed or approved by Columbia State Community College.

This page was edited on 15-Nov-2007