The next example shows how to rationalize (remove all radicals from) the denominator in an expression containing radicals.

EXAMPLE 3

Rationalizing the Denominator

Simplify each of the following expressions by rationalizing the denominator.

Solution

To rationalize the denominator, multiply by (or 1) so that the denominator of the product is a rational number.

Solution

Here, we need a perfect cube under the radical sign to rationalize the denominator. Multiplying by gives

Solution

The best approach here is to multiply both numerator and denominator by the number . The expressions are conjugates. (If a and b are real numbers, the conjugate of a + b is a - b.) Thus,

Sometimes it is advantageous to rationalize the numerator of a rational expression. The following example arises in calculus when evaluating a limit.

EXAMPLE 4

Rationalizing the Numerator

Rationalize the numerator.

Solution

Multiply numerator and denominator by the conjugate of the numerator, .

Solution

Multiply the numerator and denominator by the conjugate of the numerator,

When simplifying a square root, keep in mind that is positive by definition. Also, is not x, but the absolute value of , defined as

For example,

EXAMPLE 5

Simplifying by Factoring

Simplify .

Solution

Factor the polynomial as . Then by property 2 of radicals, and the definition of absolute value,

CAUTION

Avoid the common error of writing We must add before taking the square root. For example, This idea applies as well to higher roots. For example, in general,