Answer
$\dfrac{\sqrt[3]{150ab^2c}}{5a}$
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{\sqrt[3]{12ab^3c^2}}{\sqrt[3]{10a^3bc}}
,$ is equivalent to
\begin{align*}
&
=\dfrac{\sqrt[3]{12ab^3c^2}}{\sqrt[3]{10a^3bc}}\cdot\dfrac{\sqrt[3]{10^2b^2c^2}}{\sqrt[3]{10^2b^2c^2}}
\\\\&=
\dfrac{\sqrt[3]{12ab^3c^2(10^2b^2c^2)}}{\sqrt[3]{10^3a^3b^3c^3}}
\\\\&=
\dfrac{\sqrt[3]{1200ab^5c^4}}{\sqrt[3]{(10abc)^3}}
\\\\&=
\dfrac{\sqrt[3]{1200ab^5c^4}}{10abc}
.\end{align*}
Extracting the factors that are perfect powers of the index, the expression above is equivalent to
\begin{align*}\require{cancel}
&
=\dfrac{\sqrt[3]{8b^3c^3\cdot150ab^2c}}{10abc}
\\\\&=
\dfrac{\sqrt[3]{(2bc)^3\cdot150ab^2c}}{10abc}
\\\\&=
\dfrac{2bc\sqrt[3]{150ab^2c}}{10abc}
\\\\&=
\dfrac{\cancel2\cancel{bc}\sqrt[3]{150ab^2c}}{\cancel{10}^5a\cancel{bc}}
\\\\&=
\dfrac{\sqrt[3]{150ab^2c}}{5a}
.\end{align*}
Hence, the simplified form of the given expression is $
\dfrac{\sqrt[3]{150ab^2c}}{5a}
$.
