Question

find all cube roots of -8 answer in a+bi form

Answer 1

In polar form
\[ -8 = 8e^{i\pi} \]
and
\[ -8^{\frac{1}{3}}= 8^{\frac{1}{3}}e^{\frac{i\pi}{3}} = 2e^{\frac{i\pi}{3}} \]
Use
\[ Ae^{iB}= Acos(B)+iAsin(B) \]
to change to rectangular form.

To get the other roots (for a total of 3), multiply
-8 by 1 as in \( 8e^{i\pi} \cdot e^{i2\pi} \) and then raise to the 1/3.
Each time you multiply by e^i2pi you will get another root, until you return to your first value.