Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 9.2 - 1 Chapter 9 Roots, Radicals, and Root Functions Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 9.2 - 2 9.2 Rational Exponents Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 9.2 - 3 9.2 Rational Exponents Objectives 1. Use exponential notation for nth roots. 2. Define and use expressions of the form am/n. 3. Convert between radicals and rational exponents. 4. Use the rules for exponents with rational exponents. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 4 9.2 Rational Exponents Exponents of the Form a1/n a1/n If n a is a real number, then a1/n = Copyright © 2010 Pearson Education, Inc. All rights reserved. n a . Sec 9.2 - 5 9.2 Rational Exponents EXAMPLE 1 Evaluating Exponentials of the Form a1/n Evaluate each expression. (a) 271/3 = (b) 3 27 = 3 641/2 = 64 = 8 (c) –6251/4 = – (d) (–625)1/4 = 4 4 625 = –5 –625 is not a real number because the radicand, –625, is negative and the index is even. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 6 9.2 Rational Exponents Caution on Roots CAUTION Notice the difference between parts (c) and (d) in Example 1. The radical in part (c) is the negative fourth root of a positive number, while the radical in part (d) is the principal fourth root of a negative number, which is not a real number. EXAMPLE 1 (c) –6251/4 = (d) (–625)1/4 – = 4 4 625 = –5 –625 is not a real number because the radicand, –625, is negative and the index is even. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 7 9.2 Rational Exponents EXAMPLE 1 Evaluating Exponentials of the Form a1/n Evaluate each expression. (e) (f) (–243)1/5 = 4 25 1/2 = 5 –243 4 25 = –3 = 2 5 Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 8 9.2 Rational Exponents Exponents of the Form am/n am/n If m and n are positive integers with m/n in lowest terms, then am/n = ( a1/n ) m, provided that a1/n is a real number. If a1/n is not a real number, then am/n is not a real number. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 9 9.2 Rational Exponents EXAMPLE 2 Evaluating Exponentials of the Form am/n Evaluate each exponential. (a) 253/2 = ( 251/2 )3 = 53 = 125 (b) 322/5 = ( 321/5 )2 = 22 = 4 (c) –274/3 = –( 27)4/3 = –( 271/3 )4 = –(3)4 = –81 (d) (–64)2/3 = [(–64)1/3 ]2 = (–4)2 = 16 (e) (–16)3/2 is not a real number, since (–16)1/2 is not a real number. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 10 9.2 Rational Exponents EXAMPLE 3 Evaluating Exponentials with Negative Rational Exponents Evaluate each exponential. (a) 32–4/5 By the definition of a negative exponent, 32–4/5 = Since 324/5 = 5 4 32 = 24 1 . 4/5 32 = 16, = 32–4/5 = Copyright © 2010 Pearson Education, Inc. All rights reserved. 1 324/5 = 1 . 16 Sec 9.2 - 11 9.2 Rational Exponents EXAMPLE 3 Evaluating Exponentials with Negative Rational Exponents Evaluate each exponential. (b) 8 27 –4/3 = 1 8 27 4/3 We could also use the rule 8 27 –4/3 = 27 8 4/3 1 8 3 27 = = b a –m a b = 27 3 8 Copyright © 2010 Pearson Education, Inc. All rights reserved. 4 4 = m = 1 2 3 4 = 1 16 81 = 81 16 here, as follows. 3 2 4 = 81 16 Sec 9.2 - 12 9.2 Rational Exponents Caution on Roots CAUTION When using the rule in Example 3 (b), we take the reciprocal only of the base, not the exponent. Also, be careful to distinguish between exponential expressions like –321/5, 32–1/5, and –32–1/5. –321/5 = –2, 32–1/5 = 1 , 2 Copyright © 2010 Pearson Education, Inc. All rights reserved. and –32–1/5 = – 1 . 2 Sec 9.2 - 13 9.2 Rational Exponents Alternative Definition of am/n am/n If all indicated roots are real numbers, then am/n = ( a1/n ) m = ( a m ) 1/n. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 14 9.2 Rational Exponents Radical Form of am/n Radical Form of am/n If all indicated roots are real numbers, then am/n = n am = ( n a ) m . In words, raise a to the mth power and then take the nth root, or take the nth root of a and then raise to the mth power. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 15 9.2 Rational Exponents EXAMPLE 4 Converting between Rational Exponents and Radicals Write each exponential as a radical. Assume that all variables represent positive real numbers. Use the definition that takes the root first. 151/2 = (c) 4n2/3 = 4( 3 n )2 (d) 7h3/4 – (2h)2/5 = 7( 4 h )3 – ( 5 2h )2 (e) g–4/5 = 15 1 g4/5 (b) = 105/6 = ( 6 10 )5 (a) 1 ( 5 g )4 Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 16 9.2 Rational Exponents EXAMPLE 4 Converting between Rational Exponents and Radicals In (f) – (h), write each radical as an exponential. Simplify. Assume that all variables represent positive real numbers. (f) 33 = 331/2 (g) 3 76 = 76/3 (h) 5 m5 = m, since m is positive. = 72 = Copyright © 2010 Pearson Education, Inc. All rights reserved. 49 Sec 9.2 - 17 9.2 Rational Exponents Rules for Rational Exponents Rules for Rational Exponents Let r and s be rational numbers. For all real numbers a and b for which the indicated expressions exist: ar ( · ar ) as s = = ar + s ar s a–r ( ab ) r ar = ar – s as 1 = r a = ar b r Copyright © 2010 Pearson Education, Inc. All rights reserved. a b r ar = r b a b –r a–r br = r a = 1 a r . Sec 9.2 - 18 9.2 Rational Exponents EXAMPLE 5 Applying Rules for Rational Exponents Write with only positive exponents. Assume that all variables represent positive real numbers. (a) 63/4 · 61/2 = 63/4 + 1/2 = 65/4 (b) 32/3 35/6 = 32/3 – 5/6 = 3–1/6 = Product rule 1 31/6 Copyright © 2010 Pearson Education, Inc. All rights reserved. Quotient rule Sec 9.2 - 19 9.2 Rational Exponents EXAMPLE 5 Applying Rules for Rational Exponents Write with only positive exponents. Assume that all variables represent positive real numbers. (c) m1/4 n–6 m–8 n2/3 –3/4 = (m1/4)–3/4 (n–6)–3/4 ( m–8)–3/4 (n2/3)–3/4 = m–3/16 n9/2 m6 n–1/2 = m–3/16 – 6 n9/2 – (–1/2) = m–99/16 n5 = n5 m99/16 Copyright © 2010 Pearson Education, Inc. All rights reserved. Power rule Quotient rule Definition of negative exponent Sec 9.2 - 20 9.2 Rational Exponents EXAMPLE 5 Applying Rules for Rational Exponents Write with only positive exponents. Assume that all variables represent positive real numbers. (d) x3/5(x–1/2 – x3/4) = x3/5 · x–1/2 – x3/5 · x3/4 = x3/5 + (–1/2) – x3/5 + 3/4 Distributive property Product rule = x1/10 – x27/20 Do not make the common mistake of multiplying exponents in the first step. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 21 9.2 Rational Exponents Caution on Converting Expressions to Radical Form CAUTION Use the rules of exponents in problems like those in Example 5. Do not convert the expressions to radical form. Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 22 9.2 Rational Exponents EXAMPLE 6 Applying Rules for Rational Exponents Rewrite all radicals as exponentials, and then apply the rules for rational exponents. Leave answers in exponential form. Assume that all variables represent positive real numbers. (a) 4 a3 · 3 a2 = a3/4 · a2/3 Convert to rational exponents. = a3/4 + 2/3 Product rule = a9/12 + 8/12 Write exponents with a common denominator = a17/12 Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 23 9.2 Rational Exponents EXAMPLE 6 Applying Rules for Rational Exponents Rewrite all radicals as exponentials, and then apply the rules for rational exponents. Leave answers in exponential form. Assume that all variables represent positive real numbers. 4 (b) c = c1/4 c3/2 Convert to rational exponents. = c1/4 – 3/2 Quotient rule = c1/4 – 6/4 Write exponents with a common denominator = c–5/4 c3 = 1 c5/4 Definition of negative exponent Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 24 9.2 Rational Exponents EXAMPLE 6 Applying Rules for Rational Exponents Rewrite all radicals as exponentials, and then apply the rules for rational exponents. Leave answers in exponential form. Assume that all variables represent positive real numbers. (c) 5 3 x2 = = = 5 x2/3 ( x2/3 )1/5 x2/15 Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 25