Answer
$\frac{1}{3}$
Work Step by Step
For any positive real number $b$ (such that $b\ne1$), we know that $log_{b}b^{r}=r$ (as long as $r\gt0$).
For $log_{9}\sqrt[3] 9$:
We know that $9^{\frac{1}{3}}=\sqrt[3] 9$, so we can replace $\sqrt[3] 9$ with $9^{\frac{1}{3}}$.
Therefore, $log_{9}\sqrt[3] 9=log_{9}9^{\frac{1}{3}}=\frac{1}{3}$.