Euler's Identity

Question

Can anyone explain where did I actually commit a mistake? In the end I got the result that e^(2\pi )=1 (which is clearly not true)

Answer 1

The trick comes when you take the non-integer power of a negative number or a complex number with a non-zero imaginary part as Kavi mentioned. A contradiction involving exponents

Answer 2

You can refer to the following paradox: $$(1) ^{0.5}=(-1\times -1) ^{0.5}=(-1) ^{0.5}\times (-1) ^{0.5}=i\times i=-1$$. (Since $x^y$ is not well defined for $x, y\in\mathbb{C}$