## Intermediate Algebra (12th Edition)

$8-\sqrt{15}$
$\bf{\text{Solution Outline:}}$ 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}