Answer
$2\sqrt{15}-3\sqrt{22}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Use the Distributive Property and the properties of radicals to simplify the given expression, $
\sqrt{6}(\sqrt{10}-\sqrt{33})
.$
$\bf{\text{Solution Details:}}$
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}(\sqrt{10}-\sqrt{33})
\\\\=
\sqrt{6}(\sqrt{10})+\sqrt{6}(-\sqrt{33})
.\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}
\sqrt{6}(\sqrt{10})+\sqrt{6}(-\sqrt{33})
\\\\=
\sqrt{6(10)}-\sqrt{6(33)}
\\\\=
\sqrt{60}-\sqrt{198}
.\end{array}
Extracting the root of the factor that is a perfect power of the index results to
\begin{array}{l}\require{cancel}
\sqrt{60}-\sqrt{198}
\\\\=
\sqrt{4\cdot15}-\sqrt{9\cdot22}
\\\\=
\sqrt{(2)^2\cdot15}-\sqrt{(3)^2\cdot22}
\\\\=
2\sqrt{15}-3\sqrt{22}
.\end{array}
