If [tex]z^4 = -3 + 4j[/tex], where [tex]j = \sqrt{-1}[/tex], we first write [tex]z^4[/tex] in exponential form using
[tex]\left|z^4\right| = \sqrt{(-3)^2 + 4^2} = \sqrt{25} = 5[/tex]
and
[tex]\arg\left(z^4\right) = \pi + \tan^{-1}\left(-\dfrac43\right) = \pi - \tan^{-1}\left(\dfrac43\right)[/tex]
Then
[tex]z^4 = 5 \exp\left(j \left(\pi - \tan^{-1}\left(\dfrac43\right)\right)[/tex]
By de Moivre's theorem, the fourth roots of [tex]z^4[/tex] are
[tex]z = 5^{1/4} \exp\left(j \dfrac{\pi - \tan^{-1}\left(\frac43\right) + 2k\pi}4\right)[/tex]
with [tex]k\in\{0,1,2,3\}[/tex], so the four possible values of [tex]z[/tex] are
[tex]k = 0 \implies z_1 = 5^{1/4} \exp\left(j \dfrac{\pi - \tan^{-1}\left(\frac43\right)}4\right)[/tex]
[tex]k = 1 \implies z_2 = 5^{1/4} \exp\left(j \dfrac{3\pi - \tan^{-1}\left(\frac43\right)}4\right)[/tex]
[tex]k = 2 \implies z_3 = 5^{1/4} \exp\left(j \dfrac{5\pi - \tan^{-1}\left(\frac43\right)}4\right)[/tex]
[tex]k = 3 \implies z_4 = 5^{1/4} \exp\left(j \dfrac{7\pi - \tan^{-1}\left(\frac43\right)}4\right)[/tex]
(see attached plot from WolframAlpha)
