The problem asks us to evaluate the expression $(\frac{8}{125})^{\frac{4}{3}}$.

1. Problem Description

The problem asks us to evaluate the expression $(\frac{8}{125})^{\frac{4}{3}}$.

2. Solution Steps

First, we can rewrite the expression as:

$(\frac{8}{125})^{\frac{4}{3}} = ((\frac{8}{125})^{\frac{1}{3}})^4$.

Then we can find the cube root of $\frac{8}{125}$. Since $8=2^3$ and $125=5^3$, we have

$(\frac{8}{125})^{\frac{1}{3}} = \frac{8^{\frac{1}{3}}}{125^{\frac{1}{3}}} = \frac{2}{5}$.

Now, we can substitute this back into our original expression:

$((\frac{8}{125})^{\frac{1}{3}})^4 = (\frac{2}{5})^4$.

Therefore, we have $(\frac{2}{5})^4 = \frac{2^4}{5^4} = \frac{16}{625}$.

3. Final Answer

$\frac{16}{625}$