CPM Homework Help : PC3 Problem 12-104

Solve: $z^{3} + 1 = −i$ .

Step 1:

$z^3 = -1 - i$

Step 2:

$z=(-1-i)^{1/3}$

Step 3:

$r=\sqrt2, \theta =\frac{5\pi}{4}$

Step 4:

$z = \sqrt{2}^{1/3}\Big(\cos\Big(\frac{5\pi/4+2\pi k}{3}\Big)+ i \sin \Big(\frac{5\pi/4+2\pi k}{3}\Big) \Big)$ , for $k = 0, 1, 2$