I know that a number will have 2 square roots, 3 cube roots, and so on. This seems easy to extend to rational roots: e.g. There will be 3 2/3 roots.
But what about irrational roots?
How many roots are there to e.g. $3^{1/\pi}$? Is there a general solution?
There are always at most countably many values for such expressions, because they depend on the values of the logarithm. The possible values here are
$$z_k =e^{\frac1{\pi}(\ln 3 +2ki\pi)} = e^{\frac1{\pi}\ln 3}e^{2ki}=3^{1/\pi}\cos 2k + i3^{1/\pi}\sin 2k$$
Here, $\ln$ denotes the real-valued function of a positive real number.
The principal value is recovered with $k=0$, i.e., just the real number $3^{1/\pi}$.