Answer
$20\sqrt{5}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
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}
