## Intermediate Algebra (12th Edition)

$\dfrac{6}{r^{14}}$
Use the laws of exponents to simplify the given expression, $\dfrac{36r^{-4}(r^2)^{-3}}{6r^4} .$ Using the law of exponents which states that $(a^x)^z=a^{xz},$ the expression above simplifies to \begin{array}{l}\require{cancel} \dfrac{36r^{-4}(r^{2(-3)})}{6r^4} \\\\= \dfrac{36r^{-4}r^{-6}}{6r^4} .\end{array} Using the law of exponents which states that $a^x\cdot a^y=a^{x+y},$ the expression above simplifies to \begin{array}{l}\require{cancel} \dfrac{36r^{-4+(-6)}}{6r^4} \\\\= \dfrac{36r^{-10}}{6r^4} .\end{array} Using the law of exponents which states that $\dfrac{a^x}{a^y}=a^{x-y},$ the expression above simplifies to \begin{array}{l}\require{cancel} \dfrac{36r^{-10-4}}{6} \\\\= \dfrac{36r^{-14}}{6} .\end{array} Using the law of exponents which states that $a^{-x}=\dfrac{1}{a^x}$ or $\dfrac{1}{a^{-x}}=a^x$ the expression above simplifies to \begin{array}{l}\require{cancel} \dfrac{36}{6r^{14}} \\\\= \dfrac{6}{r^{14}} .\end{array}