Is there any way to relate this to something to the power of $y$ e.g.?
Imagine I can only do exponentiation with integer exponents. If $y$ is an integer, I cannot do $\exp$ with $\frac1y$. Is there any way to transform the expression to $\exp$ with $y$? Maybe even just an approximation?
The expression $x^{1/y}$ (or its equivalent $\sqrt[y]{x}$) just means a value $z$ that satisfies equation $z^y = x$.
Plug it in to the root-finding algorithm of your choice to get an approximation for $z$.