1. Slide 7- 1 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

2. Exponents and Radicals 7.1Radical Expressions and Functions 7.2Rational Numbers as Exponents 7.3Multiplying Radical Expressions 7.4Dividing Radical Expressions 7.5Expressions Containing Several Radical Terms 7.6Solving Radical Equations 7.7Geometric Applications 7.8 The Complex Numbers 7

3. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Dividing Radical Expressions Dividing and Simplifying Rationalizing Denominators and Numerators (Part I) 7.4

4. Slide 7- 4 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Dividing and Simplifying Just as the root of a product can be expressed as the product of two roots, the root of a quotient can be expressed as the quotient of two roots.

5. Slide 7- 5 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The Quotient Rule for Radicals For any real numbers Remember that an nth root is simplified when its radicand has no factors that are perfect nth powers. Recall too that we assume that no radicands represent negative quantities raised to an even power.

6. Slide 7- 6 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Simplify by taking roots of the numerator and denominator: Solution Taking the square roots of the numerator and denominator

7. Slide 7- 7 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution continued

8. Slide 7- 8 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Solution Divide and, if possible, simplify. Because the indices match, we can divide the radicands.

9. Slide 7- 9 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution continued

10. Slide 7- 10 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Rationalizing Denominators and Numerators (Part 1) When a radical expression appears in a denominator, it can be useful to find an equivalent expression in which the denominator no longer contains a radical. The procedure for finding such an expression is called rationalizing the denominator. We carry this out by multiplying by 1 in either of two ways.

11. Slide 7- 11 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley One way is to multiply by 1 under the radical to make the denominator of the radicand a perfect power.

12. Slide 7- 12 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Solution Rationalize each denominator. Multiplying by 1 under the radical

13. Slide 7- 13 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution Since the index is 3, we need 3 identical factors in the denominator.

14. Slide 7- 14 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Another way to rationalize a denominator is to multiply by 1 outside the radical.

15. Slide 7- 15 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Solution Rationalize each denominator. Multiplying by 1

16. Slide 7- 16 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution

17. Slide 7- 17 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Sometimes in calculus it is necessary to rationalize a numerator. To do so, we multiply by 1 to make the radicand in the numerator a perfect power.