## College Algebra (11th Edition)

$-\dfrac{2}{x^4}$
$\bf{\text{Solution Outline:}}$ Use the laws of exponents to simplify the given expression, $\dfrac{(-8xy)y^3}{4x^5y^4} .$ $\bf{\text{Solution Details:}}$ 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{-8xy^{1+3}}{4x^5y^4} \\\\= \dfrac{-8xy^{4}}{4x^5y^4} \\\\= \dfrac{\cancel4(-2)x\cancel{y^{4}}}{\cancel4x^5\cancel{y^{4}}} \\\\= \dfrac{-2x}{x^5} .\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} -2x^{1-5} \\\\= -2x^{-4} .\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{-2}{x^4} \\\\= -\dfrac{2}{x^4} .\end{array}