solve in terms of complex numbers

Question

I need the full solution (with steps) of $K^4=-4$. First, I tried to solve in termes of $K^2$ and I tried to include in my answer the j term of complex numbers.

Thanks

Answer 1

Here we are: $$ -4=4\cdot(-1)=4e^{i(\pi+2k\pi)}\;,\;k\in\mathbb Z\;. $$ so $$ z^4=-4\Leftrightarrow z^4=4e^{i(\pi+2k\pi)} $$ from which you get the solution: $$ z_k=\sqrt{2}e^{i\frac\pi{4}+ik\frac\pi{4}}. $$ Observing that as $k\in\mathbb Z$, the different solutions will be only four, so you can consider $z_k$ for $k=0,1,2,3$.