Answer
$\dfrac{2}{3}$
Work Step by Step
Note that $y=\log_a x \text{ is equivalent to } x= a^y$
Thus, if $y = \log_{5} \sqrt[3]{25}$, then $5^y=\sqrt[3]{25}$.
Since $\sqrt[3]{25}=25^{\frac{1}{3}}=(5^2)^{\frac{1}{3}}=5^{\frac{2}{3}},$ then
$5^y=5^{\frac{2}{3}}$
Use the rule $a^m=a^n \implies m=n$ to obtain:
$y=\dfrac{2}{3}$
Therefore,
$ \log_{5} \sqrt[3]{25} = \boxed{\dfrac{2}{3}}$
