Help with complex numbers

Question

Hello i'm getting ambiguity with the expression 1^i. 1 can be written as e^0 or e^(2pi*i) this leads to confusion because solving (e^0)^i, I obtain 1 and I simplify e^(2pi*i)^i and obtain e^(-2pi) which are different. Can anyone explain this?

Answer 1

We actually have that e^{-2πi}=cos(-2π)+isin(-2π)=cos(2π)-isin(2π)=1 so the answer is the same.