## Intermediate Algebra (12th Edition)

$20\sqrt{5}$
$\bf{\text{Solution Outline:}}$ To simplify the given radical expression, $2\sqrt{5}+3\sqrt{20}+4\sqrt{45} ,$ 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} 2\sqrt{5}+3\sqrt{4\cdot5}+4\sqrt{9\cdot5} \\\\= 2\sqrt{5}+3\sqrt{(2)^2\cdot5}+4\sqrt{(3)^2\cdot5} .\end{array} Extracting the roots of the factor that is a perfect power of the index results to \begin{array}{l}\require{cancel} 2\sqrt{5}+3(2)\sqrt{5}+4(3)\sqrt{5} \\\\= 2\sqrt{5}+6\sqrt{5}+12\sqrt{5} .\end{array} By combining like radicals, the expression above is equivalent to \begin{array}{l}\require{cancel} (2+6+12)\sqrt{5} \\\\= 20\sqrt{5} .\end{array}