Problem solving complex equation with high power I have to solve :$$(i+z)^{16}-(7+7i)(i+z)^8+25i=0$$
I consider this equation like a quadratic equation and find its delta and solutions. But i can't solve two equations :$$(i+z)^8=4+3i$$
$$(i+z)^8=3+4i$$
I really need some helps to solve completely this equation. Thanks.

If you don't want to use polar coordinates, the eighth roots can be found by repeated square root.
We get the square root
$$  (x+iy)^2 = a+ib $$
when
$$  x = \frac{1}{2} \left( \sqrt{ \sqrt{a^2+b^2} + b}  + \sqrt{\sqrt{a^2+b^2} - b} \right)  $$
$$  y = \frac{1}{2} \left( \sqrt{ \sqrt{a^2+b^2} + b}  - \sqrt{\sqrt{a^2+b^2} - b} \right)  $$
if the real parts $a,b$ are positive, so are the $x,y$ above are also positive. Switching $a,b$ to $b,a$ gives a different result.
