College Algebra (11th Edition)

Published by Pearson
ISBN 10: 0321671791
ISBN 13: 978-0-32167-179-0

Chapter 4 - Section 4.2 - Exponential Functions - 4.2 Exercises - Page 410: 84


$x=\{ -4,4 \}$

Work Step by Step

$\bf{\text{Solution Outline:}}$ To solve the given equation, $ x^{-4}=\dfrac{1}{256} ,$ use the laws of exponents and the properties of equality to isolate the variable. $\bf{\text{Solution Details:}}$ Using the Negative Exponent Rule of the laws of exponents which states that $x^{-m}=\dfrac{1}{x^m}$ or $\dfrac{1}{x^{-m}}=x^m,$ the expression above is equivalent to \begin{array}{l}\require{cancel} \dfrac{1}{x^4}=\dfrac{1}{256} .\end{array} Since $\dfrac{a}{b}=\dfrac{c}{d}$ implies $ad=bc$ or sometimes referred to as cross-multiplication, the equation above is equivalent to \begin{array}{l}\require{cancel} 1(256)=x^4(1) \\\\ 256=x^4 \\\\ x^4=256 .\end{array} Taking the $4th$ root of both sides, the equation above is equivalent to \begin{array}{l}\require{cancel} \sqrt[4]{x^4}=\pm\sqrt[4]{256} \\\\ x=\pm\sqrt[4]{(4)^4} \\\\ x=\pm4 .\end{array} Hence, $ x=\{ -4,4 \} .$
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