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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 Solution Here, we need a perfect cube under the radical sign to
rationalize the denominator. Multiplying by Solution The best approach here is to multiply both numerator and
denominator by the number 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 For example, EXAMPLE 5 Simplifying by Factoring Simplify Solution Factor the polynomial as CAUTION Avoid the common error of writing |