Answer
$6\sqrt[]{2}-4\sqrt[]{6}$
Work Step by Step
Using $a(b+c)=ab+ac$, or the Distributive Property, and the properties of radicals, the given expression, $
\sqrt[]{12}(\sqrt[]{6}-\sqrt[]{8})
,$ simplifies to
\begin{array}{l}\require{cancel}
\sqrt[]{12}(\sqrt[]{6})-\sqrt[]{12}(\sqrt[]{8})
\\\\=
\sqrt[]{12(6)}-\sqrt[]{12(8)}
\\\\=
\sqrt[]{(2\cdot6)(6)}-\sqrt[]{(4\cdot3)(4\cdot2)}
\\\\=
\sqrt[]{6^2\cdot2}-\sqrt[]{4^2\cdot6}
\\\\=
6\sqrt[]{2}-4\sqrt[]{6}
.\end{array}
