## Intermediate Algebra (12th Edition)

$q^{-3} \left( -5+8q \right)$
$\bf{\text{Solution Outline:}}$ Factor the variable with the lesser exponent in the given expression, $-5q^{-3}+8q^{-2} .$ Then, divide the given expression and the variable with the lesser exponent. $\bf{\text{Solution Details:}}$ Factoring $q^{-3}$ (the variable with the lesser exponent), the expression above is equivalent to \begin{array}{l}\require{cancel} q^{-3} \left( \dfrac{-5q^{-3}}{q^{-3}}+\dfrac{8q^{-2}}{q^{-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} q^{-3} \left( -5q^{-3-(-3)}+8q^{-2-(-3)} \right) \\\\= q^{-3} \left( -5q^{-3+3}+8q^{-2+3} \right) \\\\= q^{-3} \left( -5q^{0}+8q^{1} \right) \\\\= q^{-3} \left( -5(1)+8q \right) \\\\= q^{-3} \left( -5+8q \right) .\end{array}