Answer
$5a^2b^2\sqrt{5ab}+5ab\sqrt[3]{ab}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To simplify the given radical expression, $
\sqrt{125a^5b^5}+\sqrt[3]{125a^4b^4}
,$ simplify first each term by expressing the radicand as a factor that is a perfect power of the index. Then, extract the root.
$\bf{\text{Solution Details:}}$
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}
\sqrt{25a^4b^4\cdot5ab}+\sqrt[3]{125a^3b^3\cdot ab}
\\\\=
\sqrt{(5a^2b^2)^2\cdot5ab}+\sqrt[3]{(5ab)^3\cdot ab}
.\end{array}
Extracting the roots of the factor that is a perfect power of the index results to
\begin{array}{l}\require{cancel}
5a^2b^2\sqrt{5ab}+5ab\sqrt[3]{ab}
.\end{array}