Log of a Power

Let $x = \sqrt[3]{4/25}$. There is a unique value of $y$ such that $0 < y < x$ and $x^x = y^y$. What is the value of $y^3$?

(Hint: Let $k\in (0,1)$ be the ratio $y/x$, and plug in $y = kx$.)

Source: Math Prize for Girls