Answer to Question #149873 in Complex Analysis for Dhruv bartwal

Question

Find all the 5th roots of (5-i).

Answer 1

$5-i=\sqrt{26}e^{i\phi}$ , $\phi=\sin^{-1}({-\frac{1}{\sqrt{26}}})$

if $z^5=5-i$ then

$z=26^{\frac{1}{10}}e^{\frac{i\phi}{5}+\frac{2k\pi}{5}} , k=0, 1, 2, 3, 4$.