$z^5\: =\: i$

$z^5\: =\: 0\: +\: 1i$

$z^5\: =\: \sqrt{0^2\: +\: 1^2}\: (\: \cos(2k\pi\: +\frac{\: \pi}{2})\: +\: isin(2k\pi\: +\frac{\: \pi}{2}))$

$z^5\: =\: \: \cos(2k\pi+\frac{\pi}{2})+isin(2k\pi+\frac{\pi}{2})$

$z\: =\: \: \cos(\frac{2k\pi}{5}+\frac{\pi}{10})+isin(\frac{2k\pi}{5}+\frac{\pi}{10})$

$k=0\: \: \: \mapsto\: \: \: \: z_1\: =\: \cos(\frac{\pi}{10})+isin(\frac{\pi}{10})$

$k=1\: \: \: \mapsto\: \: \: \: z_2\: =\: \cos(\frac{5\pi}{10})+isin(\frac{5\pi}{10})$

$k=2\: \: \: \mapsto\: \: \: \: z_3\: =\: \cos(\frac{9\pi}{10})+isin(\frac{9\pi}{10})$

$k=3\: \: \: \mapsto\: \: \: \: z_4\: =\: \cos(\frac{13\pi}{10})+isin(\frac{13\pi}{10})$

$k=4\: \: \: \mapsto\: \: \: \: z_5\: =\: \cos(\frac{17\pi}{10})+isin(\frac{17\pi}{10})$