## Intermediate Algebra (12th Edition)

$4p-36\sqrt{p}+\sqrt{7p}-9\sqrt{7}$
$\bf{\text{Solution Outline:}}$ To simplify the given radical expression, $(4\sqrt{p}+\sqrt{7})(\sqrt{p}-9) ,$ use FOIL and the laws 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} 4\sqrt{p}(\sqrt{p})+4\sqrt{p}(-9)+\sqrt{7}(\sqrt{p})+\sqrt{7}(-9) \\\\= 4(\sqrt{p})^2+4(-9)\sqrt{p}+\sqrt{7}(\sqrt{p})+1(-9)\sqrt{7} \\\\= 4p-36\sqrt{p}+\sqrt{7}(\sqrt{p})-9\sqrt{7} .\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} 4p-36\sqrt{p}+\sqrt{7(p)}-9\sqrt{7} \\\\= 4p-36\sqrt{p}+\sqrt{7p}-9\sqrt{7} .\end{array}