## Intermediate Algebra (12th Edition)

$3\sqrt{6}+2\sqrt{3}$
$\bf{\text{Solution Outline:}}$ 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}