Answer
$3\sqrt[]{2}+2\sqrt[]{6}+6+4\sqrt{3}$
Work Step by Step
Using the FOIL method and the properties of radicals, the given expression, $
(\sqrt[]{3}+\sqrt{6})(\sqrt[]{6}+\sqrt{8})
,$ simplifies to
\begin{array}{l}\require{cancel}
\sqrt[]{3}(\sqrt[]{6})+\sqrt[]{3}(\sqrt{8})+\sqrt{6}(\sqrt[]{6})+\sqrt{6}(\sqrt{8})
\\\\=
\sqrt[]{18}+\sqrt[]{24}+6+\sqrt{48}
\\\\=
\sqrt[]{9\cdot2}+\sqrt[]{4\cdot6}+6+\sqrt{16\cdot3}
\\\\=
\sqrt[]{(3)^2\cdot2}+\sqrt[]{(2)^2\cdot6}+6+\sqrt{(4)^2\cdot3}
\\\\=
3\sqrt[]{2}+2\sqrt[]{6}+6+4\sqrt{3}
.\end{array}
