Answer
$\dfrac{\sqrt{6x}}{3x}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Use the laws of radicals to simplify the given expression, $ \sqrt{\dfrac{2}{3x}} .$ Then rationalize the denominator.
$\bf{\text{Solution Details:}}$
Using the Quotient Rule of radicals which is given by $\sqrt[n]{\dfrac{x}{y}}=\dfrac{\sqrt[n]{x}}{\sqrt[n]{y}}{},$ the expression above is equivalent to \begin{array}{l}\require{cancel} \dfrac{\sqrt{2}}{\sqrt{3x}} .\end{array}
Using the Product Rule of radicals which is given by $\sqrt[m]{x}\cdot\sqrt[m]{y}=\sqrt[m]{xy}$ to rationalize the denominator, the expression above is equivalent to\begin{array}{l}\require{cancel}
\dfrac{\sqrt{2}}{\sqrt{3x}}\cdot\dfrac{\sqrt{3x}}{\sqrt{3x}}
\\\\=
\dfrac{\sqrt{2(3x)}}{(\sqrt{3x})^2} \\\\= \dfrac{\sqrt{6x}}{3x}
.\end{array}
