Find $\displaystyle Re(z)$ and $\displaystyle Im(z)$ if:

$\displaystyle z=\sqrt[3]{ \frac{2}{-\sqrt{2+i\sqrt{2}}}}$

My idea was to simplify the expression to the normal form $\displaystyle x+iy$ (The problem is that i don't know how). And then, by De Moivre's formula I can find the roots.

Any idea on how to simplify the formula and reach the normal form?

Thank you!