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 - Page 330: 75


$p^{-3} \left( 3+2p \right)$

Work Step by Step

$\bf{\text{Solution Outline:}}$ Factor the variable with the lesser exponent in the given expression, $ 3p^{-3}+2p^{-2} .$ Then, divide the given expression and the variable with the lesser exponent. $\bf{\text{Solution Details:}}$ Factoring $ p^{-3} $ (the variable with the lesser exponent), the expression above is equivalent to \begin{array}{l}\require{cancel} p^{-3} \left( \dfrac{3p^{-3}}{p^{-3}}+\dfrac{2p^{-2}}{p^{-3}} \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} p^{-3} \left( 3p^{-3-(-3)}+2p^{-2-(-3)} \right) \\\\= p^{-3} \left( 3p^{-3+3}+2p^{-2+3} \right) \\\\= p^{-3} \left( 3p^{0}+2p^{1} \right) \\\\= p^{-3} \left( 3(1)+2p \right) \\\\= p^{-3} \left( 3+2p \right) .\end{array}
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