Section 1: Exponents and Roots

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762 Chapter 8 Exponential and Logarithmic Functionsso( )2 1 2 12 = 2 2 ·2 = 2 1 = 2,2 1 2 = ±√2.However, the negative choice for the exponent 1/2 leads to problems, because thencertain expressions are not defined. For example, it would follow from the third lawthat(2 1 2 )12 = −√− √ 2.But − √ 2 is negative, so √ − √ 2 is not defined. Therefore, it only makes sense to usethe positive choice. Thus, for all n, even and odd, 2 1/n is defined by the formula2 1 n =n √ 2.In a similar manner, for a general positive rational m n, the third law implies thatBut also,Thus,2 m n = (2 m ) 1 n =n √ 2 m .2 m n = (21n ) m = ( n√ 2) m .2 m n =n √ 2 m = ( n√ 2) m .Finally, negative rational exponents are defined in the usual manner for negativeexponents:2 − m n =12 m nMore generally, here is the final general definition. With this definition, the Lawsof Exponents hold for all rational exponents.Definition 22. For a positive rational exponent m n, and b > 0,For a negative rational exponent − m n ,b m n =n √ b m = ( n√ b) m . (23)b − m 1n = . (24)b m nRemark 25. For b < 0, the same definitions make sense only when n is odd. Forexample (−2) 1 4 is not defined.Version: Fall 2007
Section 8.1 Exponents and Roots 763◮ Example 26. Compute the exact values of (a) 4 5 2 , (b) 64 2 3 , and (c) 81 − 3 4 .a) 4 5 2 =(4 1 2) 5= (√4) 5 = 2 5 = 32b) 64 2 3 =(64 1 3) 2= (3 √ 64) 2 = 4 2 = 16c) 81 − 3 4 =181 3 4=1( )81 1 3=41( 4√ 81) = 1 3 3 3 = 127◮ Example 27. Simplify the following expressions, and write them in the form x r :a) x 2 1 23 x 4 = x 3 + 1 84 = x 12 + 3 1112 = x 12a) x 2 3 x14 , b)x 2 3x 1 4, c)( )x − 2 143b) x 2 3x 1 4c)(x − 2 3= x 2 3 − 1 4 = x812 − 312 = x512) 14= x − 2 3 · 14 = x − 212 = x − 1 6◮ Example 28.radical.Use rational exponents to simplify 5√ √ x, and write it as a single√5 √x √ )1= ( x) 5 =(x 1 152= x 1 2 · 15 = x110 =10 √ x◮ Example 29. Use a calculator to approximate 2 5/8 .Figure 8. 2 5/8 ≈ 1.542210825Version: Fall 2007
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Section 1: Exponents and Roots

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