## Intermediate Algebra (12th Edition)

$\dfrac{1}{x^{3/2}}$
$\bf{\text{Solution Outline:}}$ Use the definition of rational exponents and the laws of exponents to simplify the given expression, $\dfrac{\sqrt{x^5}}{\sqrt{x^8}} .$ $\bf{\text{Solution Details:}}$ Using the definition of rational exponents which is given by $a^{\frac{m}{n}}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m,$ the expression above is equivalent to \begin{array}{l}\require{cancel} \dfrac{x^{5/2}}{x^{8/2}} .\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} x^{\frac{5}{2}-\frac{8}{2}} \\\\ x^{-\frac{3}{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{1}{x^{\frac{3}{2}}} \\\\= \dfrac{1}{x^{3/2}} .\end{array}