# Rationalizing the Denominator

Square roots such as are irrational numbers. If roots of this type appear in the denominator of a fraction, it is customary to rewrite the fraction with a rational number in the denominator, or rationalize it. We rationalize a denominator by multiplying both the numerator and denominator by another radical that makes the denominator rational.

You can find products of radicals in two ways. By definition, is the positive number that you multiply by itself to get 2. So

By the product rule, Note that by the product rule, but By definition of a cube root,

Example 1

Rationalizing the denominator

Rewrite each expression with a rational denominator.

Solution

a) Because , multiplying both the numerator and denominator by will rationalize the denominator:

 By the product rule,

b) We must build up the denominator to be the cube root of a perfect cube. So we multiply by to get

Caution

To rationalize a denominator with a single square root, you simply multiply by that square root. If the denominator has a cube root, you build the denominator to a cube root of a perfect cube, as in Example 1(b). For a fourth root you build to a fourth root of a perfect fourth power, and so on.