College Algebra (11th Edition)

$\dfrac{5}{x^2}$
$\bf{\text{Solution Outline:}}$ Use the laws of exponents to simplify the given expression, $\dfrac{(5x)^{-2}(5x^3)^{-3}}{(5^{-2}x^{-3})^3} .$ $\bf{\text{Solution Details:}}$ Using the extended Power Rule of the laws of exponents which is given by $\left( x^my^n \right)^p=x^{mp}y^{np},$ the expression above is equivalent to \begin{array}{l}\require{cancel} \dfrac{5^{-2}x^{-2}5^{-3}x^{3(-3)}}{5^{-2(3)}x^{-3(3)}} \\\\= \dfrac{5^{-2}x^{-2}5^{-3}x^{-9}}{5^{-6}x^{-9}} \\\\= \dfrac{5^{-2}x^{-2}5^{-3}\cancel{x^{-9}}}{5^{-6}\cancel{x^{-9}}} \\\\= \dfrac{5^{-2}x^{-2}5^{-3}}{5^{-6}} .\end{array} 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{5^{-2+(-3)}x^{-2}}{5^{-6}} \\\\= \dfrac{5^{-2-3}x^{-2}}{5^{-6}} \\\\= \dfrac{5^{-5}x^{-2}}{5^{-6}} .\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} 5^{-5-(-6)}x^{-2} \\\\= 5^{-5+6}x^{-2} \\\\= 5^{1}x^{-2} \\\\= 5x^{-2} .\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{5}{x^2} .\end{array}