Express both sides of the equation with the same base.

We can express both 125 and 25 as powers of 5.

$$125 = 5^3$$ $$25 = 5^2$$

Substitute these into the equation:

$$(5^3)^y = \frac{1}{5^2}$$

Recall the exponent rule $(a^m)^n = a^{mn}$.

$$5^{3y} = \frac{1}{5^2}$$

Recall the exponent rule $\frac{1}{a^m} = a^{-m}$.

$$5^{3y} = 5^{-2}$$

Since the bases are the same, the exponents must be equal.

$$3y = -2$$

Divide both sides by 3:

$$y = -\frac{2}{3}$$

The value of $y$ is:

$$-\frac{2}{3}$$