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?

By the definition used in the context of complex numbers, the multivaled expression $z^b = \exp(b \log(z))$ where $\log(z)$ is any branch of the logarithm.  In this case $\log(1) = 2 n \pi i$ for an integer $i$, so $1^b = \exp(2 n b \pi i)$.  If $b$ is irrational, these are all distinct, so there are infinitely many values.
