what are all the possible roots

  {8, -4 +i·4√3, -4 -i·4√3}

Step-by-step explanation:

Using Euler's formula, we can find the roots from ...

  512 = 8^3 = 8^3·e^(2nπi)

Then the cube root is ...

  root = 512^(1/3) = 8e^(2nπi/3)

The values will be different for n = 0, 1, 2; then they repeat. So the three cube roots of 512 are ...

The latter two can be written as the complex numbers ...

  8e^(2/3πi) = 8(cos(2/3π) +i·sin(2/3π)) = -4 +i·4√3

and

  8e^(4/3πi) = 8(cos(4/3π) +i·sin(4/3π)) = -4 -i·4√3

Then the three cube roots of 512 are ...

  {8, -4 +i·4√3, -4 -i·4√3}

These are evenly spaced around the circle with radius 8 in the complex plane.