I'm preparing for an exam and I'm facing some troubles with complex numbers, so any help would be much much appreciated! I've been re-reading the chapter like 10 times by now and I just can't figure it out.
Find the solutions of the following equations: (a) $z4 = −8 + 8√3i$
from here our $r_0$ is $^4\sqrt16=2$
$z^4=16(-\frac12+\frac{\sqrt3}2i)$
Now trying to find a1 and b1 for z1:
$a1=(r_0*cos\beta):n=-1$
$b1=(r_0*sin\beta):n=\sqrt3$
Hence, $z_1 = -1 + \sqrt3i$
However, in the textbook, the value is different, it gives for an answer for $z1=z_1 = \sqrt3 -1i$ So I don't know what am I doing wrong or how to continue finding z2, z3 and z4 from here on.
(b) $z3 + \frac58i = \frac{15}i$
Here I get to a point where $z3= 125:8i$ and im pretty much lost from here on. Any help would be highly appreciated! Thank you in advance for your time!
\begin{align}z^4&=-8+8\sqrt{3}i\\&=16\left(-\frac12+\frac{\sqrt3}2i\right)\\ &=16\left( \cos\left( \frac{2\pi}3 \right) + i\sin\left( \frac{2\pi}3 \right)\right)\\ &=2^4\exp\left(i\left( \frac{2\pi}3+2k\pi\right) \right)\end{align}
where $k=0,1,2,3$.
Try to take it from here.