Question

Find all of the fifth roots of 32i?

Answer 1

$z^5=32i=32e^{i\pi/2}$

$\implies z=32^{1/5}e^{i(\pi/2+2n\pi)/5}$

where $n=0,1,2,3,4$. So the fifth roots of $32i$, given by each possible $z$, are

$z_1=2e^{i\pi/10}$
$z_2=2e^{i(\pi/10+2\pi/5)}=2e^{i\pi/2}$
$z_3=2e^{i(\pi/10+4\pi/5)}=2e^{9i\pi/10}$
$z_4=2e^{i(\pi/10+6\pi/5)}=2e^{13i\pi/10}$
$z_5=2e^{i(\pi/10+8\pi/5)}=2e^{17i\pi/10}$