What if in certain algebra $w^{-n}$ is multivalued?

Question

In complex numbers we have $z^{1/n}$ and $a^z$ having multiple values.

But what would happen if the inverse function $1/z$ were multi-valued?

For instance, there is such $w$ that $1/w=\pi^2/6$ but $w\ne 6/\pi^2$.