Answer
$8-\sqrt{15}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given radical expression, $
(2\sqrt{3}+\sqrt{5})(3\sqrt{3}-2\sqrt{5})
,$ use FOIL and the properties of radicals. Then combine like terms.
$\bf{\text{Solution Details:}}$
Using the FOIL Method which is given by $(a+b)(c+d)=ac+ad+bc+bd,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
2\sqrt{3}(3\sqrt{3})-2\sqrt{3}(2\sqrt{5})+\sqrt{5}(3\sqrt{3})+\sqrt{5}(-2\sqrt{5})
\\\\=
2(3)(\sqrt{3})^2-2(2)\sqrt{3}(\sqrt{5})+1(3)\sqrt{5}(\sqrt{3})+1(-2)(\sqrt{5})^2
\\\\=
6(3)-4\sqrt{3}(\sqrt{5})+3\sqrt{5}(\sqrt{3})-2(5)
\\\\=
18-4\sqrt{3}(\sqrt{5})+3\sqrt{5}(\sqrt{3})-10
.\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}
18-4\sqrt{3(5)}+3\sqrt{5(3)}-10
\\\\=
18-4\sqrt{15}+3\sqrt{15}-10
.\end{array}
By combining like terms, the expression above is equivalent to
\begin{array}{l}\require{cancel}
18-4\sqrt{3(5)}+3\sqrt{5(3)}-10
\\\\=
(18-10)+(-4\sqrt{15}+3\sqrt{15})
\\\\=
8-\sqrt{15}
.\end{array}
