I figured out that if $x^{y} = z$, then $z^y = x^{y^2}$. Then we know Euler's Formula:
$$e^{πi} = -1,\quad (e^π)^i = -1,\quad (e^{2π})^i = 1$$
Now, using the formula above, let $e^{2π}$ act as x, and let i act as y. Then, finally, let 1 act as z. Accordingly, $z^{y}$, which is $1^i$, equals $x^{y^2}$, which is $(e^{2π})^{i^2}$. This means:
$$1^i = (e^{2π})^{-1}$$
Since $1^i = 1$:
$$1 = 1/(e^{2π})$$ $$(e^{2π}) = 1$$
But $(e^{2π}) ≠ 1$, so...(!!)
How is this possible?! And I think I actually figured this out!!!
Exponents only behave well under certain conditions. For instance, when the base is a positive real number and the exponent is real, or when the base is non-zero and the exponent is an integer.
The more you move away from such requirements, the more careful you have to be when applying rules like $a^{b+c}=a^ba^c$ and $a^{bc}=(a^b)^c$. They will not always work.