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.
Best Answer
In general $$x^3+a^3=(x+a)(x^2-ax+a^2)$$
You can apply this to get a linear term and a quadratic term in your problem. Find the roots of the quadratic term to get the other two roots.