Copyright © 2008 Pearson Education, Inc Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Rationalizing the Denominator 8.4 Rationalizing the Denominator Rationalize denominators with square roots. Write radicals in simplified form. Rationalize denominators with cube roots. 1 2 3 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Rationalize denominators with square roots. Objective 1 Rationalize denominators with square roots. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8.4 - 3

Rationalize denominators with square roots. It is easier to work with a radical expression if the denominators do not contain any radicals. This process of changing the denominator from a radical, or irrational number, to a rational number is called rationalizing the denominator. The value of the radical expression is not changed; only the form is changed, because the expression has been multiplied by 1 in the form of Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8.4 - 4

EXAMPLE 1 Rationalize each denominator. Solution: Rationalizing Denominators EXAMPLE 1 Rationalize each denominator. Solution: Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8.4 - 5

Write radicals in simplified form. Objective 2 Write radicals in simplified form. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8.4 - 6

Write radicals in simplified form. A radical is considered to be in simplified form if the following three conditions are met. The radicand contains no factor (except 1) that is a perfect square (in dealing with square roots), a perfect cube (in dealing with cube roots), and so on. The radicand has no fractions. No denominator contains a radical. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8.4 - 7

EXAMPLE 2 Solution: Simplifying a Radical Slide 8.4 - 8 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8.4 - 8

EXAMPLE 3 Simplify Solution: Simplifying a Product of Radicals Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8.4 - 9

EXAMPLE 4 Simplify . Assume that p and q are positive numbers. Simplifying a Quotient of Radicals Simplify . Assume that p and q are positive numbers. Solution: Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8.4 - 10

EXAMPLE 5 Simplify . Assume that r and t represent Simplifying a Radical Quotient EXAMPLE 5 Simplify . Assume that r and t represent nonnegative real numbers. Solution: Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8.4 - 11

Rationalize denominators with cube roots. Objective 3 Rationalize denominators with cube roots. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8.4 - 12

EXAMPLE 6 Rationalize each denominator. Solution: Rationalizing Denominators with Cube Roots Rationalize each denominator. Solution: Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8.4 - 13