Answer
$-\dfrac{10}{3}$ and $\dfrac{10}{3}$
Work Step by Step
Note that
$\left(\frac{10}{3}\right)^4=\frac{10,000}{81}$ and
$\left(-\frac{10}{3}\right)^4=\frac{10,000}{81}$.
Thus, the real fourth roots of $\frac{10,000}{81}$ are $-\frac{10}{3}$ and $\frac{10}{3}$.