Answer
$\text{radicand: }
a(a+b)
\\\text{index: }
6$
Work Step by Step
The radicand and the index of the given expression, $
\dfrac{a^2}{b}\sqrt[6]{a(a+b)}
,$ are
\begin{array}{l}
\text{radicand: }
a(a+b)
\\\text{index: }
6
.\end{array}
Recall that in the expression $\sqrt[y] x$, y is the index while x is the radicand.