Originally Posted by
ThePerfectHacker 1)$\displaystyle \sqrt{3} - i = | \sqrt{3} - i | e^{i \arg (\sqrt{3} - i)} = 2 e^{-i\pi/6}$.
Therefore, the fourth roots are: $\displaystyle 2^{1/4} e^{-\pi i/24}, -2^{1/4} e^{-\pi i/24}, i2^{1/4}e^{-\pi i/24}, -i 2^{1/4} e^{-\pi i/24}$.
2)$\displaystyle -i = e^{-\pi i /2}$.
Therefore, the fifth roots are: $\displaystyle e^{-\pi i/10}, \zeta e^{-\pi i/10}, \zeta^2 e^{-\pi i/10}, \zeta^3 e^{-\pi i/10}, \zeta^4 e^{-\pi i/10}$.
Where $\displaystyle \zeta = e^{2\pi i/5}$