The complex number $z = 4^{\frac{1}{6}} \left(-\frac{\sqrt{3}}{2} + \frac{i}{2}\right)$ satisfies the equation $z^6$ = -4

Question

Quizplus · Accepted Answer

The complex number given is in the form $r(\cos \theta + i\sin \theta)$ where $r = 4^{\frac{1}{6}}$ and $\theta = \frac{5\pi}{6}$, which corresponds to the polar form of a complex number. Raising this number to the 6th power multiplies the magnitude $r^6 = 4$ and raises the angle to $6 \times \frac{5\pi}{6} = 5\pi$, resulting in a complex number that points in the negative real direction with a magnitude of 4, which is $-4$. This satisfies the equation $z^6 = -4$.

The Complex Number z=416(−32+i2)z = 4^{\frac{1}{6}} \left(-\frac{\sqrt{3}}{2} + \frac{i}{2}\right)z=461​(−23​​+2i​) Satisfies the Equation

The Complex Number $z = 4^{\frac{1}{6}} \left(-\frac{\sqrt{3}}{2} + \frac{i}{2}\right)$ Satisfies the Equation