## Intermediate Algebra (12th Edition)

$-49p\sqrt{3}$
$\bf{\text{Solution Outline:}}$ To simplify the given radical expression, $9\sqrt{27p^2}-14\sqrt{108p^2}+2\sqrt{48p^2} ,$ simplify first each term by expressing the radicand as a factor that is a perfect power of the index. Then, extract the root. Finally, combine the like radicals. $\bf{\text{Solution Details:}}$ Expressing the radicand as an expression that contains a factor that is a perfect power of the index results to \begin{array}{l}\require{cancel} 9\sqrt{9p^2\cdot3}-14\sqrt{36p^2\cdot3}+2\sqrt{16p^2\cdot3} \\\\= 9\sqrt{(3p)^2\cdot3}-14\sqrt{(6p)^2\cdot3}+2\sqrt{(4p)^2\cdot3} .\end{array} Extracting the roots of the factor that is a perfect power of the index results to \begin{array}{l}\require{cancel} 9(3p)\sqrt{3}-14(6p)\sqrt{3}+2(4p)\sqrt{3} \\\\= 27p\sqrt{3}-84p\sqrt{3}+8p\sqrt{3} .\end{array} By combining like radicals, the expression above is equivalent to \begin{array}{l}\require{cancel} (27p-84p+8p)\sqrt{3} \\\\= -49p\sqrt{3} .\end{array}