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 Radical Expressions and Functions 1.Find the nth root of a number. 2.Approximate roots using a calculator. 3.Simplify radical expressions. 4.Evaluate radical functions. 8.1

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 4 Evaluating nth roots When evaluating a radical expression, the sign of a and the index n will determine possible outcomes. If a is nonnegative, then, where and b n = a. If a is negative and n is even, then there is no real- number root. If a is negative and n is odd, then, where b is negative and b n = a. nth root: The number b is an nth root of a number a if b n = a.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 5 Example Evaluate each root, if possible. a. b. c. Solution is not a real number because there is no real number whose square is –100.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 6 continued Evaluate each root, if possible. d. e. f. Solution

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 7 continued Evaluate each root, if possible. g. h. Solution

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 8 Some roots, like are called irrational because we cannot express their exact value using rational numbers. In fact, writing with the radical sign is the only way we can express its exact value. However, we can approximate using rational numbers. Approximating to two decimal places: Approximating to three decimal places: Note: Remember that the symbol,, means “approximately equal to.”

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 9 Example Approximate the roots using a calculator or table in the endpapers. Round to three decimal places. a. b. c. Solution

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 10 Example Find the root. Assume variables represent nonnegative values. a. b. c. Solution Because (y 2 ) 2 = y 4. Because (6m 3 ) 2 = 36m 6. Solution

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 11 continued Find the root. Assume variables represent nonnegative values. d. e. Solution

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 12 Example Find the root. Assume variables represent any real number. a. b. c. Solution

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 13 continued Find the root. Assume variables represent any real number. d. e. c. Solution

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 14 Radical function: A function containing a radical expression whose radicand has a variable. Example Solution Given f(x) = find f(3). To find f(3), substitute 3 for x and simplify.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 15 Example Find the domain of each of the following. a. b. Solution Since the index is even, the radicand must be nonnegative. Solution The radicand must be nonnegative. Domain: 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. 16 Example If you drop an object, the time (t) it takes in seconds to fall d feet is given by. 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, replacing d with 800. Execute Replace d with 800. Divide within the radical. Evaluate the square root.

Copyright © 2015, 2011, 2007 Pearson Education, Inc. 17 continued Answer It takes an object seconds to fall 800 feet. Check We can verify the calculations, which we will leave to the viewer.