Solve $x^4 + 64 = 0$

Question

I know this question has been solved in here:

Factor $x^4 + 64$

But I solved in another way and I don't seem to be getting the same solution and I don't know where is the error:

Here is my attempt:

$x^4 + 64 = 0$
$x^4 = -64$
Let $m = x^2$
$m^2 = -64,$ take the square root on both sides
$m_1= +i8 , m_2 = -i8$, substitute x back:
$x^2 = i8$, take the square root on both sides
$x^2 = -i8$, take the square root on both sides
We get:
$x_1 = + \sqrt{i8}= 2 \sqrt{2}\sqrt{i}$,
$x_2 = + \sqrt{i8}= 2 \sqrt{2}\sqrt{i}$,
$x_3 = - \sqrt{-i8}= 2i \sqrt{2}\sqrt{i}$,
$x_4 = - \sqrt{-i8}= 2i \sqrt{2}\sqrt{i}$,

When I plug the equation in Wolf alpha it gave this result: $x = \pm (2 - 2 i)$, $x = \pm (2 - 2 i)$

Where is the mistake in my solution?

Answer 1

$i=\cos(\pi/2)+i\sin(\pi/2)$, so $\sqrt{i}=\cos(\pi/4)+i\sin(\pi/4)=\sqrt{2}(i+1)/2$ or $\cos(5\pi/4)+i\sin(5\pi/4)=-\sqrt{2}(i+1)/2$ depends on which branch you choose.

Do similar thing on $-i$, plug into your $\sqrt{i}$ and $\sqrt{-i}$, your answer should be the same as the one you get from Wolf alpha.