## Intermediate Algebra (12th Edition)

$p^{-3} \left( 3+2p \right)$
$\bf{\text{Solution Outline:}}$ 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}