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

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

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

Copyright © 2010 Pearson Education, Inc. All rights reserved. Rational Exponents Caution on Roots (c) –625 1/4 (d) (–625) 1/4 = EXAMPLE 1 =

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

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

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec Evaluate each exponential. (a) 25 3/2 EXAMPLE 2 (b) 32 2/5 (c) –27 4/3 (d) (–64) 2/3 (e) (–16) 3/2

Evaluate each exponential. (a) 32 –4/5 EXAMPLE 3 Evaluating Exponentials with Negative Rational Exponents

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

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

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

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

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

Copyright © 2010 Pearson Education, Inc. All rights reserved. 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 Rational Exponents (f)33 =33 1/2 =7 6/3 (g) =7272 =49 = m, since m is positive. (h) m5m5 5

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

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 (b) 3 2/3 3 5/6

Copyright © 2010 Pearson Education, Inc. All rights reserved. Write with only positive exponents. Assume that all variables represent positive real numbers. EXAMPLE 5 (c) m 1/4 n –6 m –8 n 2/3 –3/4

Copyright © 2010 Pearson Education, Inc. All rights reserved. Write with only positive exponents. Assume that all variables represent positive real numbers. EXAMPLE 5 Rational Exponents (d) x 3/5 ( x –1/2 – x 3/4 ) Do not make the common mistake of multiplying exponents in the first step.

Copyright © 2010 Pearson Education, Inc. All rights reserved. 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. 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 Rational Exponents (a)· 4 a3a3 3 a2a2

Copyright © 2010 Pearson Education, Inc. All rights reserved. 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 Rational Exponents (b) 4 c c3c3