Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

Published by Pearson
ISBN 10: 0-32184-874-8
ISBN 13: 978-0-32184-874-1

Chapter 4 - Polynomials - 4.2 Negative Exponents and Scientific Notations - 4.2 Exercise Set: 125

Answer

$(-17)^{-8}$ is positive because when the negative exponent rule is applied, the denominator's power is even (which is $8$). Any non-zero real number raised to an even power is positive. Refer to the step-by-step part below for a detailed explanation.

Work Step by Step

RECALL: For any nonzero real number $a$, $a^m$ is positive when $m$ is an even integer. The reason for this is that an even power $m$ means there are $\frac{m}{2}$ pairs of identical factors. Since the product of two numbers with the same sign is positive, then having $\frac{m}{2}$ pairs of identical factors means that the product will surely be positive. Note that $a^{-m} = \dfrac{1}{a^m}, m \ne 0$. Applying this to $(-17)^{-8}$ gives $\dfrac{1}{(-17)^8}$. Since the denominator involves an even power of $-17$, then the denominator is positive. Thus, the quotient will also be positive as the quotient of two positive numbers is positive.
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