Why is this wrong (complex numbers and proving 1=-1)?

Question

$$(e^{2πi})^{1/2}=1^{1/2}$$$$(e^{πi})=1$$ $$-1=1$$ I think it is due to not taking the principle value but please can someone explain why this is wrong in detial, thanks.

Answer 1

You used two different branches of the function $x^{\frac{1}{2}}$.

Note that even in exponential form $(e^{x})^\frac{1}{2}$ has two different branches: $e^{\frac{x}{2}}$ and $e^{\frac{x}{2}+\frac{2 \pi i}{2}}$.

Answer 2

You basically claimed

$$ \sqrt{(-1)^2} = \sqrt{(1)^2} \quad \Rightarrow \quad -1 = 1 $$

but the square root is not a map, i.e. $\sqrt{a}$ can have multiple solutions. In particular, it is not the inverse of $x^2$.