Intermediate Algebra (12th Edition)

Published by Pearson
ISBN 10: 0321969359
ISBN 13: 978-0-32196-935-4

Chapter 5 - Section 5.1 - Greatest Common Factors and Factoring by Grouping - 5.1 Exercises: 74

Answer

$k^{-4} \left( k^{2}+2 \right)$

Work Step by Step

$\bf{\text{Solution Outline:}}$ Factor the variable with the lesser exponent in the given expression, $ k^{-2}+2k^{-4} .$ Then, divide the given expression and the variable with the lesser exponent. $\bf{\text{Solution Details:}}$ Factoring $ k^{-4} $ (the variable with the lesser exponent), the expression above is equivalent to \begin{array}{l}\require{cancel} k^{-4} \left( \dfrac{k^{-2}}{k^{-4}}+\dfrac{2k^{-4}}{k^{-4}} \right) .\end{array} Using the Quotient Rule of the laws of exponents which states that $\dfrac{x^m}{x^n}=x^{m-n},$ the expression above simplifies to \begin{array}{l}\require{cancel} k^{-4} \left( k^{-2-(-4)}+2k^{-4-(-4)} \right) \\\\= k^{-4} \left( k^{-2+4}+2k^{-4+4} \right) \\\\= k^{-4} \left( k^{2}+2k^{0} \right) \\\\= k^{-4} \left( k^{2}+2(1) \right) \\\\= k^{-4} \left( k^{2}+2 \right) .\end{array}
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