Answer
$\dfrac{1}{x^{3/2}}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
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}
