Assume $Z_1$ and $Z_2$ are two complex numbers such that $Z_1 = a+ib$ and $Z_2 = c+id$. When I do the exponent operation $Z_1^{Z_2}$ I get infinitely many answers generally, since writing $Z_1$ in polar form, I can have many arguments solving them, each separated by $2 \pi$ radians, so I write it as a series of principal argument + $2n \pi$ ; where $n$ is set of integers. (The solution can be unique in certain scenarios, where $b=0$, that is, $Z_2$ is purely real.)
Assuming $c$ and $d$ to be integers I get many solutions all having same principal argument but varying magnitudes; I'm keeping $c$ and $d$ are completely integers to avoid certain illegal solutions (square root of $e^{ia}$ and value of $e^{ia/2}$ may not be the same).
The problem I'm facing is If I'm replacing $a=e$ and $b=0$ (that is, writing $e^{c+id}$) in this form I again get infinite solutions with different magnitudes, all with same principal argument.
One of them is what we get if using Euler's formula, will have magnitude $e^c$ if $n=0$.
Now I'm confused if there is a problem with the general expression with infinite solutions making them invalid, or if it is correct and the Euler's solution is one of many possible solutions?
Expanding on the same, is $1/Z_1^{Z_2} = (1/Z_1)^{Z_2}$? I am assuming it won't be since $1$ and $1^{Z_2}$ are not the same. But if I write it as $1 \cdot Z_1^{-Z2} = Z_1^{-Z_2} = (1/Z_1)^{Z_2}$ it satisfies.
One of the above is wrong and I'm not sure which.