College Algebra (11th Edition)

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

Chapter 4 - Review Exercises: 35

Answer

$x=\dfrac{22}{5}$

Work Step by Step

$\bf{\text{Solution Outline:}}$ To solve the given equation, $ 16^{x+4}=8^{3x-2} ,$ use the laws of exponents to expressed both sides in the same base. Then equate the exponents. Use the properties of equality to isolate the variable. $\bf{\text{Solution Details:}}$ Using exponents, the given equation is equivalent to \begin{array}{l}\require{cancel} (2^4)^{x+4}=(2^3)^{3x-2} .\end{array} Using the Power Rule of the laws of exponents which is given by $\left( x^m \right)^p=x^{mp},$ the equation above is equivalent to \begin{array}{l}\require{cancel} 2^{4(x+4)}=2^{3(3x-2)} .\end{array} Since the bases are the same, then the exponents can be equated. Hence, \begin{array}{l}\require{cancel} 4(x+4)=3(3x-2) \\\\ 4(x)+4(4)=3(3x)+3(-2) \\\\ 4x+16=9x-6 \\\\ 4x-9x=-6-16 \\\\ -5x=-22 \\\\ x=\dfrac{-22}{-5} \\\\ x=\dfrac{22}{5} .\end{array}
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