Answer
$\dfrac{15|x|\sqrt{2x}}{2}$
Work Step by Step
Using the Quotient Rule of radicals which is given by $\sqrt[n]{\dfrac{x}{y}}=\dfrac{\sqrt[n]{x}}{\sqrt[n]{y}}{},$ the given expression, $
\dfrac{3\sqrt{100x^2}}{2\sqrt{2x^{-1}}}
,$ is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{3}{2}\sqrt{\dfrac{100x^2}{2x^{-1}}}
\\=
\dfrac{3}{2}\sqrt{\dfrac{50x^2}{x^{-1}}}
.\end{array}
Using the Quotient Rule of the laws of exponents which states that $\dfrac{x^m}{x^n}=x^{m-n},$ the expression above simplifies to
\begin{array}{l}\require{cancel}
\dfrac{3}{2}\sqrt{50x^{2-(-1)}}
\\=
\dfrac{3}{2}\sqrt{50x^{2+1}}
\\=
\dfrac{3}{2}\sqrt{50x^{3}}
.\end{array}
Extracting the factor of the radicand that is a perfect power of the index, the expression above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{3}{2}\sqrt{50x^{3}}
\\=
\dfrac{3}{2}\sqrt{25x^{2}\cdot2x}
\\=
\dfrac{3}{2}\sqrt{(5x)^{2}\cdot2x}
.\end{array}
Using $\sqrt[n]{x^n}=|x|$ if $n$ is even and $\sqrt[n]{x^n}=x$ if $n,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{3}{2}|5x|\sqrt{2x}
\\\\=
\dfrac{3}{2}\cdot5|x|\sqrt{2x}
\\\\=
\dfrac{15|x|\sqrt{2x}}{2}
.\end{array}