## Intermediate Algebra (12th Edition)

$\dfrac{12x^3\sqrt{2xy}}{y^{5}}$
$\bf{\text{Solution Outline:}}$ To rationalize the given radical expression, $\sqrt{\dfrac{288x^7}{y^9}} ,$ multiply the radicand by an expression equal to $1$ which will make the denominator a perfect power of the index. $\bf{\text{Solution Details:}}$ Multiplying the radicand by an expression equal to $1$ which will make the denominator a perfect power of the index results to \begin{array}{l}\require{cancel} \sqrt{\dfrac{288x^7}{y^9}\cdot\dfrac{y}{y}} \\\\= \sqrt{\dfrac{288x^7y}{y^{10}}} .\end{array} Writing the radicand as an expression that contains a factor that is a perfect power of the index results to \begin{array}{l}\require{cancel} \sqrt{\dfrac{144x^6}{y^{10}}\cdot2xy} \\\\= \sqrt{\left( \dfrac{12x^3}{y^{5}}\right)^2\cdot2xy} .\end{array} Extracting the root of the factor that is a perfect power of the index results to \begin{array}{l}\require{cancel} \dfrac{12x^3}{y^{5}}\sqrt{2xy} \\\\= \dfrac{12x^3\sqrt{2xy}}{y^{5}} .\end{array}