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Chapter 8 Rational Exponents, Radicals, and Complex Numbers Copyright © 2015, 2011, 2007 Pearson Education, Inc. 11 CHAPTER 8 Rational Exponents, Radicals, and Complex Numbers 8.1 8.2 8.3 8.4 8.5 8.6 8.7 Radical Expressions and Functions Rational Exponents Multiplying, Dividing, and Simplifying Radicals Adding, Subtracting, and Multiplying Radical Expressions Rationalizing Numerators and Denominators of Radical Expressions Radical Equations and Problem Solving Complex Numbers Copyright © 2015, 2011, 2007 Pearson Education, Inc. 2 8.1 1. 2. 3. 4. Radical Expressions and Functions Find the nth root of a number. Approximate roots using a calculator. Simplify radical expressions. Evaluate radical functions. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 3 nth root: The number b is an nth root of a number a if bn = a. Evaluating nth roots When evaluating a radical expression n a , the sign of a and the index n will determine possible outcomes. If a is nonnegative, then n a b, where b 0 and bn = a. If a is negative and n is even, then there is no realnumber root. If a is negative and n is odd, then n a b , where b is negative and bn = a. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 4 Example Evaluate each root, if possible. a. 169 Solution 169 13 b. 0.49 Solution 0.49 0.7 c. 100 Solution 100 is not a real number because there is no real number whose square is –100. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 5 continued Evaluate each root, if possible. d. 144 Solution 144 12 49 e. 144 Solution f. 3 7 49 49 12 144 144 27 Solution 3 27 3 Copyright © 2015, 2011, 2007 Pearson Education, Inc. 6 continued Evaluate each root, if possible. g. 3 27 Solution 3 27 3 h. 4 81 Solution 4 81 3 Copyright © 2015, 2011, 2007 Pearson Education, Inc. 7 Some roots, like 3 are called irrational because we cannot express their exact value using rational numbers. In fact, writing 3 with the radical sign is the only way we can express its exact value. However, we can approximate 3 using rational numbers. Approximating to two decimal places: 2 1.41 Approximating to three decimal places: 2 1.414 Note: Remember that the symbol, “approximately equal to.” , means Copyright © 2015, 2011, 2007 Pearson Education, Inc. 8 Example Approximate the roots using a calculator or table in the endpapers. Round to three decimal places. a. 18 Solution 18 4.243 b. 32 Solution 32 5.657 c. 3 56 Solution 3 56 3.826 Copyright © 2015, 2011, 2007 Pearson Education, Inc. 9 Example Find the root. Assume variables represent nonnegative values. a. y Solution 4 b. 36m 6 36 x10 c. 25 y 4 Solution Solution y4 y2 Because (y2)2 = y4. 3 6m 36m 6 Because (6m3)2 = 36m6. 36 x10 6 x5 2 4 25 y 5y Copyright © 2015, 2011, 2007 Pearson Education, Inc. 10 continued Find the root. Assume variables represent nonnegative values. d. 3 e. 4 y Solution 9 16 81x Solution 3 y9 4 y3 4 81x16 3x Copyright © 2015, 2011, 2007 Pearson Education, Inc. 11 Example Find the root. Assume variables represent any real number. Solution 14 a. y b. 36 y 10 Solution c. (n 3) Solution 2 7 y14 y 5 36 y10 6 y (n 3) 2 n 3 Copyright © 2015, 2011, 2007 Pearson Education, Inc. 12 continued Find the root. Assume variables represent any real number. 12 6 49 y 7 y 49 y Solution d. 12 e. 3 9 27n Solution 3 27n9 3n3 c. 3 ( w 4)3 Solution 3 ( w 4)3 w 4 Copyright © 2015, 2011, 2007 Pearson Education, Inc. 13 Radical function: A function containing a radical expression whose radicand has a variable. Example Given f(x) = 5 x 8, find f(3). Solution To find f(3), substitute 3 for x and simplify. f 3 5 3 8 15 8 7 Copyright © 2015, 2011, 2007 Pearson Education, Inc. 14 Example Find the domain of each of the following. a. f x x 8 Solution Since the index is even, the radicand x 8 0 x 8 must be nonnegative. Domain: x x 8 , or [8, ) b. f x 3x 9 Solution The radicand must be nonnegative. 3x 9 0 3x 9 Domain: x x 3 , or (,3] x3 Conclusion The domain of a radical function with an even index must contain values that keep its radicand nonnegative. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 15 Example If you drop an object, the time (t) it takes in seconds to fall d feet is given by t 16d . Find the time it takes for an object to fall 800 feet. Understand We are to find the time it takes for an object to fall 800 feet. Plan Use the formula t Execute t 800 16 d 16 , replacing d with 800. Replace d with 800. t 50 Divide within the radical. t 7.071 Evaluate the square root. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 16 continued Answer It takes an object 7.071 seconds to fall 800 feet. Check We can verify the calculations, which we will leave to the viewer. Copyright © 2015, 2011, 2007 Pearson Education, Inc. 17