Answer
$\dfrac{2|x|\sqrt{3x}}{5}$
Work Step by Step
Extracting the factor that is a perfet power of the index, then
\begin{array}{l}\require{cancel}
\sqrt{\dfrac{12x^3}{25}}
\\\\=
\sqrt{\dfrac{4x^2}{25}\cdot3x}
\\\\=
\sqrt{\left( \dfrac{2x}{5} \right)^2\cdot3x}
.\end{array}
Using $\sqrt[n]{x^n}=|x|$ if $n$ is even and $\sqrt[n]{x^n}=x$ if $n$ is odd, then
\begin{array}{l}\require{cancel}
\sqrt{\left( \dfrac{2x}{5} \right)^2\cdot3x}
\\\\=
\left| \dfrac{2x}{5} \right|\sqrt{3x}
\\\\=
\left| \dfrac{2}{5} \right|\cdot|x|\sqrt{3x}
\\\\=
\dfrac{2}{5}|x|\sqrt{3x}
\\\\=
\dfrac{2|x|\sqrt{3x}}{5}
.\end{array}
