Answer
$3\sqrt{6}+2\sqrt{3}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given radical expression, $
\sqrt{6}(3+\sqrt{2})
,$ use the Distributive Property and the properties of radicals.
$\bf{\text{Solution Details:}}$
Using the Distributive Property which is given by $a(b+c)=ab+ac,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
\sqrt{6}(3)+\sqrt{6}(\sqrt{2})
\\\\=
3\sqrt{6}+\sqrt{6}(\sqrt{2})
.\end{array}
Using the Product Rule of radicals which is given by $\sqrt[m]{x}\cdot\sqrt[m]{y}=\sqrt[m]{xy},$ the expression above is equivalent to\begin{array}{l}\require{cancel}
3\sqrt{6}+\sqrt{6(2)}
\\\\=
3\sqrt{6}+\sqrt{12}
.\end{array}
Rewriting the radicand with a factor that is a perfect power of the index, the given expression is equivalent to
\begin{array}{l}\require{cancel}
3\sqrt{6}+\sqrt{4\cdot3}
\\\\=
3\sqrt{6}+\sqrt{(2)^2\cdot3}
.\end{array}
Extracting the root of the factor that is a perfect power of the index results to
\begin{array}{l}\require{cancel}
3\sqrt{6}+2\sqrt{3}
.\end{array}
