## Intermediate Algebra (12th Edition)

$k^{-4} \left( k^{2}+2 \right)$
$\bf{\text{Solution Outline:}}$ 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}