## Algebra: A Combined Approach (4th Edition)

$\dfrac{2x^2\sqrt{2xy}}{y}$
$\bf{\text{Solution Outline:}}$ To rationalize the denominator of the given expression, $\sqrt{\dfrac{24x^5}{3y}} ,$ simplify first and then multiply by an expression equal to $1$ which will make the denominator a perfect power of the index. $\bf{\text{Solution Details:}}$ Simplifying the given radical expression results to \begin{array}{l}\require{cancel} \sqrt{\dfrac{8x^5}{y}} .\end{array} Multiplying the given expression by an expression equal to $1$ which will make the denominator a perfect power of the index and then simplifying the radical result to \begin{array}{l}\require{cancel} \sqrt{\dfrac{8x^5}{y}\cdot\dfrac{y}{y}} \\\\= \sqrt{\dfrac{8x^5y}{y^2}} .\end{array} Extracting the perfect root of the index results to \begin{array}{l}\require{cancel} \sqrt{\dfrac{4x^4}{y^2}\cdot2xy} \\\\= \sqrt{\left( \dfrac{2x^2}{y} \right)^2\cdot2xy} \\\\= \dfrac{2x^2}{y}\sqrt{2xy} \\\\= \dfrac{2x^2\sqrt{2xy}}{y} .\end{array}