$e^{2\sqrt{2} \pi i}=(e^{2 \pi i})^{\sqrt{2}}=1$? This is probably stupid. But this true?$$e^{2\sqrt{2} \pi i}=(e^{2 \pi i})^{\sqrt{2}}=1$$
I feel like this is wrong but I cannot see how. Any help is appreciated. Thank you

The "obvious" identity $(a^b)^c=a^{bc}$ does not hold in complete generality.
It does hold for all complex numbers $a$ if $b$ and $c$ are restricted to be integers (with a minor caveat for $a=0$ if $b$ and $c$ and negative integers).
It does hold for all positive real numbers $a$ if $b$ and $c$ are restricted to be real numbers, provided you define the exponential notation so that $x^y$ is a positive real number when $x$ is a positive real and $y$ is real, e.g., $x^y=e^{y\ln x}=\sum_{n=0}^\infty{(y\ln x)^n\over n!}$ with $\ln x=\int_1^x{dt\over t}$.
There are other circumstances under which it holds, but in general it does not.  In the problem at hand, $a=e$ is a positive real, and $c=\sqrt2$ is real, but $b=2\pi i$ is not, so you cannot expect the identity to hold.
