## Intermediate Algebra (12th Edition)

$\sqrt{39}+\sqrt{143}-\sqrt{21}-\sqrt{77}$
$\bf{\text{Solution Outline:}}$ To simplify the given radical expression, $(\sqrt{13}-\sqrt{7})(\sqrt{3}+\sqrt{11}) ,$ 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{13}(\sqrt{3})+\sqrt{13}(\sqrt{11})-\sqrt{7}(\sqrt{3})-\sqrt{7}(\sqrt{11}) .\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{13(3)}+\sqrt{13(11)}-\sqrt{7(3)}-\sqrt{7(11)} \\\\= \sqrt{39}+\sqrt{143}-\sqrt{21}-\sqrt{77} .\end{array}