Rational Exponents, Radicals, and Complex Numbers CHAPTER 10 Rational Exponents, Radicals, and Complex Numbers 10.1 Radical Expressions and Functions 10.2 Rational Exponents 10.3 Multiplying, Dividing, and Simplifying Radicals 10.4 Adding, Subtracting, and Multiplying Radical Expressions 10.5 Rationalizing Numerators and Denominators of Radical Expressions 10.6 Radical Equations and Problem Solving 10.7 Complex Numbers Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Radical Expressions and Functions 10.1 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. 5. Find the domain of radical functions. 6. Solve applications involving radical functions. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

nth root: The number b is an nth root of a number a if bn = a. 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 bn = 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 bn = a. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Evaluate each square root. a. b. Solution c. Solution Solution Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Approximating to two decimal places: 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 © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Approximate the roots using a calculator or table in the endpapers. Round to three decimal places. a. b. Solution Solution Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Find the root. Assume variables represent nonnegative values. c. Solution Because (y2)2 = y4. Solution Because (6m3)2 = 36m6. Solution Because (2x4)5 = 32x20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Find the root. Assume variables represent any real number. a. c. Solution Solution Solution Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

To find f(3), substitute 3 for x and simplify. Radical function: A function of the form f(x) = where P is a polynomial. Example Given f(x) = find f(3). Solution To find f(3), substitute 3 for x and simplify. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Find the domain of each of the following. a. b. Solution Since the index is even, the radicand must be nonnegative. Domain: 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 © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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 © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

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 © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

For which square root is –12.37 the approximation for? b) c) d) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

For which square root is –12.37 the approximation for? b) c) d) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Evaluate. a) 0.2 b) 0.02 c) 0.002 d) 0.0002 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Evaluate. a) 0.2 b) 0.02 c) 0.002 d) 0.0002 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Find the domain of f(x) = . b) c) d) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Find the domain of f(x) = . b) c) d) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

10.2 Rational Exponents 1. Evaluate rational exponents. 2. Write radicals as expressions raised to rational exponents. 3. Simplify expressions with rational number exponents using the rules of exponents. 4. Use rational exponents to simplify radical expressions. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Rational exponents: An exponent that is a fraction. Rational Exponents with a Numerator of 1 a1/n = where n is a natural number other than 1. Note: If a is negative and n is odd, then the root is negative. If a is negative and n is even, then there is no real number root. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Rewrite using radicals, then simplify. a. 491/2 b. 1251/3 c. 641/6 Solution a. b. c. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

General Rule for Rational Exponents where a  0 and m and n are natural numbers other than 1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Rewrite using radicals, then simplify, if possible. a. 272/3 b. 2433/4 c. 95/2 Solution a. b. c. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Negative Rational Exponents where a  0, and m and n are natural numbers with n  1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Rewrite using radicals, then simplify, if possible. a. 251/2 b. 272/3 Solution a. b. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Write each of the following in exponential form. a. b. Solution a. b. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Rules of Exponents Summary (Assume that no denominators are 0, that a and b are real numbers, and that m and n are integers.) Zero as an exponent: a0 = 1, where a 0. 00 is indeterminate. Negative exponents: Product rule for exponents: Quotient rule for exponents: Raising a power to a power: Raising a product to a power: Raising a quotient to a power: Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Use the rules of exponents to simplify. Write the answer with positive exponents. Solution Use the product rule for exponents. (Add the exponents.) Add the exponents. Simplify the rational exponent. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Use the rules of exponents to simplify. Write the answer with positive exponents. Solution Use the quotient for exponents. (Subtract the exponents.) Rewrite the subtraction as addition. Add the exponents. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Perform the indicated operations. Write the result using a radical. a. b. Solution a. b. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Simplify. a) b) c) d) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Simplify. a) b) c) d) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Simplify. a) 5 b) 25 c) 25 d) 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Simplify. a) 5 b) 25 c) 25 d) 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Simplify. a) 4 b) c) 4 d) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Simplify. a) 4 b) c) 4 d) Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley