How to solve $(-19w + 93\overline w)^4=-1$

Question

How to solve $(-19w + 93\overline w)^4=-1$ , if $w\in \mathbb C$

I really have no direction where to solve this question or at least a hint, can someone help?

Answer 1

HINT: Let $z=-19w+93\bar{w}$. Then $z=\dfrac{1+i}{\sqrt2}$ or $\dfrac{-1+i}{\sqrt2}$, or $\dfrac{-1-i}{\sqrt2}$, or $\dfrac{1-i}{\sqrt2}$ (roots of $-1$). Let $w=a+bi$. Then in the first case $$ -19a-19bi+93a-93bi=\dfrac{1+i}{\sqrt2} $$ and the rest should be obvious.