Answer
$\dfrac{12x^3\sqrt{2xy}}{y^{5}}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
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}