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

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

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 3 Multiplying, Dividing, and Simplifying Radicals 1.Multiply radical expressions. 2.Divide radical expressions. 3.Use the product rule to simplify radical expressions. 8.3

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 4 Product Rule for Radicals If both and are real numbers, then

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 5 Example Find the product and simplify. Assume all variables represent positive values. a.b. Solution a.b.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 6 continued Find the product and simplify. Assume all variables represent positive values. c.d. Solution c.d.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 7 continued Find the product and simplify. Assume all variables represent positive values. e.f. Solution e.f.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 8 continued Find the product and simplify. Assume all variables represent positive values. g. Solution g.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 9 Raising an nth Root to the nth Power For any nonnegative real number a, Quotient Rule for Radicals If both and are real numbers, then

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 10 Example Simplify. Assume variables represent positive values. a. Solution b. a. c.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 11 continued Simplify. Assume variables represent positive values. d. Solution e. d.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 12 Simplifying nth Roots To simplify an nth root, 1. Write the radicand as a product of the greatest possible perfect nth power and a number or an expression that has no perfect nth power factors. 2. Use the product rule when a is the perfect nth power. 3. Find the nth root of the perfect nth power radicand.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 13 Example Simplify. a.b. Solution

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 14 continued Simplify. c.d. Solution

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 15 Example Simplify the radical using prime factorization. Solution Write 686 as a product of its prime factors. The square root of the pair of 7s is 7. Multiply the prime factors in the radicand.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 16 continued Simplify the radical using prime factorization. b.c. b. Solution c.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 17 Example Simplify. Solution The greatest perfect square factor of 32x 5 is 16x 4. Use the product rule of square roots to separate the factors into two radicals. Find the square root of 16x 4 and leave 2x in the radical.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 18 Example Simplify Solution The greatest perfect square factor of 96a 4 b is 16a 4. Use the product rule of square roots to separate the factors into two radicals. Find the square root of 16a 4 and leave 6b in the radical. Multiply 2 and 4.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 19 continued Simplify. c.d. Solution

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 20 Example Find the product or quotient and simplify the results. Assume that variables represent positive values. a. b. Solution

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 21 continued Find the product or quotient and simplify the results. Assume that variables represent positive values. c. d. Solution