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Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 9.2 - 1

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Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 9.2 - 2 Roots, Radicals, and Root Functions Chapter 9

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Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 9.2 - 3 9.2 Rational Exponents

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Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 4 9.2 Rational Exponents Objectives 1.Use exponential notation for n th roots. 2.Define and use expressions of the form a m/n. 3.Convert between radicals and rational exponents. 4.Use the rules for exponents with rational exponents.

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Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 5 9.2 Rational Exponents Exponents of the Form a 1/ n a 1/ n If is a real number, then a n a 1/n =. a n

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Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 6 Evaluate each expression. (a) EXAMPLE 1 Evaluating Exponentials of the Form a 1/n 9.2 Rational Exponents 27 1/3 27 3 = 3 (b) 64 1/2 64 = 8 = = (c) –625 1/4 625 4 = –5= – (d) (–625) 1/4 –625 4 is not a real number because the radicand, –625, is negative and the index is even. =

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Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 7 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. (c) –625 1/4 625 4 = –5= – (d) (–625) 1/4 –625 4 is not a real number because the radicand, –625, is negative and the index is even. = EXAMPLE 1

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Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 8 (e) (–243) 1/5 –243 5 = –3= Evaluate each expression. EXAMPLE 1 Evaluating Exponentials of the Form a 1/n 9.2 Rational Exponents (f) 1/2 4 25 = 4 = 2 5

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Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 9 9.2 Rational Exponents Exponents of the Form a m / n am/nam/n If m and n are positive integers with m/n in lowest terms, then a m/n = ( a 1 /n ) m, provided that a 1 /n is a real number. If a 1 /n is not a real number, then a m/n is not a real number.

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Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 10 Evaluate each exponential. (a) 25 3/2 EXAMPLE 2 Evaluating Exponentials of the Form a m/n 9.2 Rational Exponents = ( 25 1/2 ) 3 = 5 3 = 125 (b) 32 2/5 = ( 32 1/5 ) 2 = 2 2 = 4 (c) –27 4/3 = –( 27 1/3 ) 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. = –( 27) 4/3

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Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 11 Evaluate each exponential. (a) 32 –4/5 EXAMPLE 3 Evaluating Exponentials with Negative Rational Exponents 9.2 Rational Exponents 1 32 4/5 By the definition of a negative exponent, 32 –4/5 =. Since 32 4/5 = 4 = 2 4 = 16, 5 32 = 32 –4/5 =. 1 32 4/5 = 1 16

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Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 12 Evaluate each exponential. (b) EXAMPLE 3 Evaluating Exponentials with Negative Rational Exponents 9.2 Rational Exponents –4/3 8 27 4/3 8 27 1 == 8 3 4 1 2 3 = 4 1 16 81 = 1 16 = We could also use the rule = here, as follows. –m–m b a m a b –4/3 8 27 4/3 27 8 = 8 3 = 4 = 4 3 2 81 16 =

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Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 13 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 –32 1/5, 32 –1/5, and –32 –1/5. –32 1/5 = –2, 1 2 32 –1/5 =, 1 2 and –32 –1/5 = –.

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Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 14 9.2 Rational Exponents Alternative Definition of a m / n am/nam/n If all indicated roots are real numbers, then a m/n = ( a 1 /n ) m = ( a m ) 1 /n.

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Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 15 9.2 Rational Exponents Radical Form of a m / n If all indicated roots are real numbers, then 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. a m/n = = ( ). n amam n a m

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Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 16 Write each exponential as a radical. Assume that all variables represent positive real numbers. Use the definition that takes the root first. (a) 15 1/2 EXAMPLE 4 Converting between Rational Exponents and Radicals 9.2 Rational Exponents 15= (b) 10 5/6 = ( ) 5 10 6 (c) 4 n 2/3 = 4( ) 2 n 3 (d) 7 h 3/4 – (2 h ) 2/5 = 7( ) 3 h 4 (e) g –4/5 = 1 g 4/5 = 1 ( ) 4 g 5 – ( ) 2 5 2h2h

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Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 17 In (f) – (h), write each radical as an exponential. Simplify. Assume that all variables represent positive real numbers. EXAMPLE 4 Converting between Rational Exponents and Radicals 9.2 Rational Exponents (f)33 =33 1/2 =7 6/3 (g)7676 3 =7272 =49 = m, since m is positive. (h) m5m5 5

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Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 18 9.2 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: a r · a s = a r + s ( a r ) s = a r s ( ab ) r = a r b r a –r = 1 arar = a r – s arar asas = a b –r brbr arar = a b r arar brbr a –r = 1 a r.

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Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 19 Write with only positive exponents. Assume that all variables represent positive real numbers. (a) 6 3/4 · 6 1/2 EXAMPLE 5 Applying Rules for Rational Exponents 9.2 Rational Exponents = 6 3/4 + 1/2 = 6 5/4 Product rule Quotient rule= 3 2/3 – 5/6 (b) 3 2/3 3 5/6 = 3 –1/6 1 3 1/6 =

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Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 20 Write with only positive exponents. Assume that all variables represent positive real numbers. EXAMPLE 5 Applying Rules for Rational Exponents 9.2 Rational Exponents Power rule (c) m 1/4 n –6 m –8 n 2/3 –3/4 = ( m –8 ) –3/4 ( n 2/3 ) –3/4 ( m 1/4 ) –3/4 ( n –6 ) –3/4 = m 6 n –1/2 m –3/16 n 9/2 = m –3/16 – 6 n 9/2 – (–1/2) Quotient rule = m –99/16 n 5 Definition of negative exponent = m 99/16 n5n5

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Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 21 Write with only positive exponents. Assume that all variables represent positive real numbers. EXAMPLE 5 Applying Rules for Rational Exponents 9.2 Rational Exponents (d) x 3/5 ( x –1/2 – x 3/4 ) = x 3/5 · x –1/2 – x 3/5 · x 3/4 Distributive property = x 3/5 + (–1/2) – x 3/5 + 3/4 Product rule = x 1/10 – x 27/20 Do not make the common mistake of multiplying exponents in the first step.

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Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 22 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.

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Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 23 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. EXAMPLE 6 Applying Rules for Rational Exponents 9.2 Rational Exponents Convert to rational exponents. (a)· 4 a3a3 3 a2a2 = a 3/4 · a 2/3 = a 3/4 + 2/3 = a 9/12 + 8/12 = a 17/12 Product rule Write exponents with a common denominator

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Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 24 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. EXAMPLE 6 Applying Rules for Rational Exponents 9.2 Rational Exponents Convert to rational exponents. Quotient rule Write exponents with a common denominator (b) 4 c c3c3 = c 1/4 c 3/2 = c 1/4 – 3/2 = c 1/4 – 6/4 = c –5/4 = c 5/4 1 Definition of negative exponent

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Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 9.2 - 25 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. EXAMPLE 6 Applying Rules for Rational Exponents 9.2 Rational Exponents (c) 3 x2x2 5 x 2/3 5 = ( x 2/3 ) 1/5 = x 2/15 =

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