Not sure how to proceed with $z$ and $i$ on the lhs -

Question

I'm not sure how to separate the $z$ and $i$ on the LHS. I know $i^2 = -1$ but I suspect there is an easier way then expanding $(z+i)$. $$ (z+i)^4 = -16. $$

Answer 1

one way would be to set $ x = (z+i) $ and solve for $x^4 = -16$. This equation has four solutions, namely $$ x_{1,2,3,4} = \pm\sqrt{2} \pm \sqrt{2}i $$

Therefore you get four solutions for z: $$ z_j = x_j - i, \quad j = 1,\dots,4. $$