Answer
$a^{5}b^{5}\sqrt{b}$
Work Step by Step
Extracting the factors that are perfect roots of the given index, the given expression, $
\sqrt{a^{10}b^{11}}
,$ simplifies to
\begin{array}{l}\require{cancel}
\sqrt{a^{10}b^{10}\cdot b}
\\\\=
\sqrt{(a^{5}b^{5})^2\cdot b}
\\\\=
a^{5}b^{5}\sqrt{b}
\end{array}
* Note that it is assumed that no radicands were formed by raising negative numbers to even powers.