## College Algebra (11th Edition)

$\dfrac{1}{a^{23}}$
$\bf{\text{Solution Outline:}}$ Use the laws of exponents to simplify the given expression, $\dfrac{a^{-6}(a^{-8})}{a^{-2}(a^{11})} .$ $\bf{\text{Solution Details:}}$ Using the Product Rule of the laws of exponents which is given by $x^m\cdot x^n=x^{m+n},$ the expression above is equivalent to \begin{array}{l}\require{cancel} \dfrac{a^{-6+(-8)}}{a^{-2+11}} \\\\= \dfrac{a^{-6-8}}{a^{-2+11}} \\\\= \dfrac{a^{-14}}{a^{9}} .\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} a^{-14-9} \\\\= a^{-23} .\end{array} Using the Negative Exponent Rule of the laws of exponents which states that $x^{-m}=\dfrac{1}{x^m}$ or $\dfrac{1}{x^{-m}}=x^m,$ the expression above is equivalent to \begin{array}{l}\require{cancel} \dfrac{1}{a^{23}} .\end{array}