## Intermediate Algebra (12th Edition)

$\sqrt{6}-\sqrt{2}+\sqrt{3}-1$
$\bf{\text{Solution Outline:}}$ To simplify the given radical expression, $(\sqrt{2}+1)(\sqrt{3}-1) ,$ use FOIL and the properties of radicals. $\bf{\text{Solution Details:}}$ 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} \sqrt{2}(\sqrt{3})+\sqrt{2}(-1)+1(\sqrt{3})+1(-1) \\\\= \sqrt{2}(\sqrt{3})-\sqrt{2}+\sqrt{3}-1 .\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{2(3)}-\sqrt{2}+\sqrt{3}-1 \\\\= \sqrt{6}-\sqrt{2}+\sqrt{3}-1 .\end{array}