Answer
$-\dfrac{\sqrt[]{33x}}{4x}$
Work Step by Step
Multiplying the numerator and the denominator by an expression equal to $1$ which will make the denominator a perfect power of the index, the given expression, $
\dfrac{3\sqrt[]{11x^3y}}{-2\sqrt[]{12x^4y}}
,$ is equivalent to
\begin{align*}\require{cancel}
&
\dfrac{3\sqrt[]{11x^3y}}{-2\sqrt[]{12x^4y}}\cdot\dfrac{\sqrt[]{3y}}{\sqrt[]{3y}}
\\\\&=
\dfrac{3\sqrt[]{33x^3y^2}}{-2\sqrt[]{36x^4y^2}}
\\\\&=
\dfrac{3\sqrt[]{x^2y^2\cdot33x}}{-2\sqrt[]{(6x^2y)^2}}
\\\\&=
\dfrac{3\sqrt[]{(xy)^2\cdot33x}}{-2\sqrt[]{(6x^2y)^2}}
\\\\&=
\dfrac{3xy\sqrt[]{33x}}{-2(6x^2y)}
\\\\&=
\dfrac{\cancel3\cancel x\cancel y\sqrt[]{33x}}{-2(\cancel6^2x^\cancel2\cancel y)}
\\\\&=
\dfrac{\sqrt[]{33x}}{-2(2x)}
\\\\&=
-\dfrac{\sqrt[]{33x}}{4x}
.\end{align*}
Hence, the simplified form of the given expression is $
-\dfrac{\sqrt[]{33x}}{4x}
$.
