how to find the imaginary roots of fourth degree cannot be simplified?

Question

How to find all roots of the following:

$$x^4 + 4 = 0$$ I know all roots will be imaginary. I tried to find a solution on the internet but could not find any.

Answer 1

The roots are $4^{1/4}\omega$ where $\omega$ is a fourth root of $-1$, namely $4^{1/4}e^{i(2k-1)\pi/4}, k=0,1,2,3$.

Answer 2

From the Sophie Germain identity, we have $x^4+4 = (x^2+2x+2)(x^2-2x+2)$, the roots of the quadratic polynomials are $-1+i,-1-i,1+i,1-i$.