Answer
$\dfrac{2\sqrt{6}}{x}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To rationalize the given radical expression, $
\sqrt{\dfrac{24}{x}}
,$ multiply both the numerator and the denominator by an expression that will make the denominator a perfect power of the index. Note that the variables are assumed to represent positive real numbers.
$\bf{\text{Solution Details:}}$
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{24}{x}\cdot\dfrac{x}{x}}
\\\\=
\sqrt{\dfrac{24x}{x^2}}
.\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{4}{x^2}\cdot6}
\\\\=
\sqrt{\left(\dfrac{2}{x}\right)^2\cdot6}
.\end{array}
Extracting the root of the factor that is a perfect power of the index results to
\begin{array}{l}\require{cancel}
\\\\=
\dfrac{2}{x}\sqrt{6}
\\\\=
\dfrac{2\sqrt{6}}{x}
.\end{array}
