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

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

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

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Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 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 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 Evaluate each expression. (a) EXAMPLE 1 Evaluating Exponentials of the Form a 1/n 9.2 Rational Exponents 27 1/ = 3 (b) 64 1/2 64 = 8 = = (c) –625 1/ = –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 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/ = –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 (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/ = 4 = 2 5

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

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

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

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Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 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 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 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 = ( ) (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 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) =7272 =49 = m, since m is positive. (h) m5m5 5

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Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 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 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 =

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