How many solutions does $1=x^π$ have? I was wondering how many solutions there are to $1 = x^\text{irrational number}$, since the cube root of 1 has 3 solutions and the 4th root has 4 etc and since the number of solutions to $x = x^{a/b}$ is b (where $a$ and $b$ share no factors), how many would $x^π=1$ have? Infinity, none or something else?

We have
$$1^\pi = (\mathrm e^{2\pi\mathrm in})^\pi = \mathrm e^{2\pi^2\mathrm in}$$
Now $2\pi^2$ is incommensurable with $2\pi$. Therefore we have a countable infinity of solutions, one for each $n\in\mathbb Z$. In particular, the solutions are dense on the unit circle.
