Let $\{{z_1,..,z_5}\}$ Solutions of $z^5=2i$
Find $x^2=\frac{1}{z_1}+...+\frac{1}{z_5}+i$
I found the solutions of $z^5=2i$ :
$z_1=\sqrt[5]{2}cis\left(\frac{\pi }{10}\right)$
$z_2=\sqrt[5]{2}cis\left(\frac{\pi }{2}\right)$
$z_3=\sqrt[5]{2}cis\left(\frac{9\pi }{10}\right)$
$z_4=\sqrt[5]{2}cis\left(\frac{13\pi }{10}\right)$
$z_5=\sqrt[5]{2}cis\left(\frac{17\pi }{10}\right)$
now I need to find what is $x^2=\frac{1}{z_1}+...+\frac{1}{z_5}+i$
and here I'm stuck How to continue from here ?
Thanks