When simplifying any expression, we try to make it look â€œsimpler.â€ When simplifying a radical expression, we have three specific conditions to satisfy.

A radical expression of index n is in simplified radical form if it has

1. no perfect nth powers as factors of the radicand,

2. no fractions inside the radical, and

3. no radicals in the denominator.

The radical expressions in the next example do not satisfy the three conditions for simplified radical form. To rewrite an expression in simplified form, we use the product rule, the quotient rule, and rationalizing the denominator.

Example 1

Simplify.

Solution

a) To rationalize the denominator, multiply the numerator and denominator by :

 Rationalize the denominator. Remove the perfect square from Reduce Note that

b) To rationalize the denominator, build up the denominator to a cube root of a perfect cube. Because , we multiply by :

 Quotient rule for radicals Rationalize the denominator.

c) To rationalize the denominator, observe that

 Quotient rule for radicals Rationalize the denominator. Do not omit the index of the radical in any step.