Answer
$-2x\sqrt[3]{3}$
Work Step by Step
Using $a(b+c)=ab+ac$, or the Distributive Property, and the properties of radicals, the given expression, $
\sqrt[3]{x}(\sqrt[3]{3x^2}-\sqrt[3]{81x^2})
,$ is equivalent to
\begin{array}{l}\require{cancel}
(\sqrt[3]{x})(\sqrt[3]{3x^2})-(\sqrt[3]{x})(\sqrt[3]{81x^2})
\\\\=
\sqrt[3]{x(3x^2)}-\sqrt[3]{x(81x^2)}
\\\\=
\sqrt[3]{3x^3}-\sqrt[3]{81x^3}
\\\\=
\sqrt[3]{x^3\cdot3}-\sqrt[3]{27x^3\cdot3}
\\\\=
\sqrt[3]{(x)^3\cdot3}-\sqrt[3]{(3x)^3\cdot3}
\\\\=
x\sqrt[3]{3}-3x\sqrt[3]{3}
\\\\=
-2x\sqrt[3]{3}
.\end{array}
