$e^{2\sqrt{2} \pi i}=(e^{2 \pi i})^{\sqrt{2}}=1$? 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

$a^{mn}=(a^m)^n$ is true only if  $a >0$ and $a \in \mathbb R$.
Here, in your question, $a=e^{2\sqrt 2 \pi i}$ isn't purely real, so you can't apply this.
Because if you go to apply this property for imaginary numbers also then weird things such as following will occur -$$i=e^{i\pi/2}=e^{2\pi I /4}=(e^{2i\pi})^{1/4}=1$$
