1. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 1 1 Chapter 8 Rational Exponents, Radicals, and Complex Numbers

2. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 2 Rational Exponents, Radicals, and Complex Numbers 8.1Radical Expressions and Functions 8.2Rational Exponents 8.3Multiplying, Dividing, and Simplifying Radicals 8.4Adding, Subtracting, and Multiplying Radical Expressions 8.5Rationalizing Numerators and Denominators of Radical Expressions 8.6Radical Equations and Problem Solving 8.7Complex Numbers CHAPTER 8

3. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 3 Rationalizing Numerators and Denominators of Radical Expressions 1.Rationalize denominators. 2.Rationalize denominators that have a sum or difference with a square root term. 8.5

4. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 4 Example Rationalize the denominator. Solution Simplify. Multiply by

5. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 5 Example Rationalize the denominator. Solution Use the quotient rule for square roots to separate the numerator and denominator into two radicals. Multiply by Simplify. Warning: Never divide out factors common to a radicand and a number not under a radical.

6. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 6 Example Rationalize the denominator. Solution

7. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 7 Rationalizing Denominators To rationalize a denominator containing a single nth root, multiply the fraction by a well chosen 1 so that the product’s denominator has a radicand that is a perfect nth power.

8. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 8 Example Rationalize the denominator. Assume that variables represent positive values. Solution

9. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 9 Example Rationalize the denominator. Assume that variables represent positive values. Solution

10. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 10 Example Rationalize the denominator. Assume that variables represent positive values. Solution

11. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 11 Rationalizing a Denominator Containing a Sum or Difference To rationalize a denominator containing a sum or difference with at least one square root term, multiply the fraction by a 1 whose numerator and denominator are the conjugate of the denominator.

12. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 12 Example Rationalize the denominator and simplify. Assume variables represent positive values. Solution

13. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 13 Example Rationalize the denominator and simplify. Assume variables represent positive values. Solution

14. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 14 Example Rationalize the denominator and simplify. Assume variables represent positive values. Solution

15. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 15 Example Rationalize the numerator. Assume variables represent positive values. Solution

16. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 16 Example Rationalize the numerator. Assume variables represent positive values. Solution