What are the cube roots of zero plus 8i?

Answer 1

The 3 cube-roots of 8i are:

1. √3 + i
2. -√3 + i
3. -2i

Think of 8i in polar form as 8∠90°. A number raised to a power (in this case 1/3) is the magnitude raised to the power and the angle is the angle times the power.

So 8 raised to 1/3 power is 2. And 90° * (1/3) = 30°. To find the angles of the other 2 cube-roots, find equivalent angles (add 360° & 720°). So you have 450°/3 = 150° and 810°/3 = 270°.

So the three roots: 2∠30°, 2∠150° & 2∠270°. Which are the three answers, above (in rectangular coordinates)