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

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

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