Imaginary number observation

Question

What is the reason for this ?

$ (i^4)^{\frac 12} = 1^{\frac 12} = 1$

$(i^4)^{\frac 12} = {({(i^2)}^2)}^{\frac 12} = i^2 = -1 $

Here, $$ i=\sqrt{-1}$$

Answer 1

The properties of the powers that you are using are not valid for complex numbers if the exponents are not integer.

Answer 2

The reason for that is that every complex number (excepto $0$) has two square roots. The notation $a^{\frac12}$ is ambiguous.

Answer 3

Because of the discontinuous nature of the square root function in the complex plane, in general it is not true that $\sqrt{zw}=\sqrt{z}\sqrt{w}$, but if $z,w$ nonnegative real numbers it becomes true. Assuming this rule you can "proof" that 1=-1: $$-1=ii=\sqrt{-1}\sqrt{-1}=(\neq)\sqrt{(-1)(-1)}=\sqrt{1}=1$$