<s>[INST] 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.
 [/INST] 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.
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