algebra precalculus - Imaginary part reduction

I have following expression
$\sqrt[3]{a + bi} + \sqrt[3]{a - bi} =$

How can I calculate it? I would like to have a solution with imaginary part reduction because Im sure that solution is a real number.

Answer

Hints: 1) Write $a+bi = r e^{i\theta}$, 2) use DeMoivre's formula. Note that if $a+bi = r e^{i\theta}$, then $a-bi = r e^{-i\theta}$, so

$$(r e^{i\theta})^{1/3} + (r e^{-i\theta})^{1/3} = r^{1/3}\left( \cos\frac{\theta}{3} + i\sin\frac{\theta}{3} + \cos\left(-\frac{\theta}{3}\right) + i\sin\left(-\frac{\theta}{3}\right) \right) = 2r^{1/3}\cos\frac{\theta}{3}.$$