complex cube roots

["barr"]

It is well known and easy to prove that square roots of complex numbers
can be calculated using real square roots.  That is you can solve the
equation (x + iy)^2 = a + ib for x and y using only real square roots.
My question is, is it possible to calculate complex cube roots using
only real cube and, if necessay, square roots.  I tried, but the
equations I got seemed intractible.