Complex solutions to $ x^3 + 512 = 0 $ An algebra book has the exercise 
$$ x^3 + 512 = 0 $$
I can find the real solution easily enough with 
$$ x^3 = -512 $$
$$ \sqrt[3]{x^3} = \sqrt[3]{-512} $$
$$ x = -8 $$
The book also gives the complex solutions $$ 4 \pm 4\sqrt{3}i $$
But I don't understand how to find these answers. Having completed the chapter on complex numbers I can find square roots of negative numbers easily, but cube (or higher) roots are never explained.

If you want to solve the complex roots using quadratic formula, try below :
$$x^3 + 512 = x^3+8^3 = (x+8)(x^2-8x+64) = 0$$
