2.5.2—de Moivre's theorem

To solve w^n=z=r(cosθ+i sinθ), the roots have modulus r^(1/n) and arguments (θ+2πk)/n for k=0,1,…,n−1.

The n values are equally spaced on a circle and are distinct modulo 2π. Give them in polar form or convert each to Cartesian form as requested.

The cube roots of 8 are modulus 2 with arguments 0, 2π/3 and 4π/3, giving 2, −1+i√3 and −1−i√3.

Using only θ/n misses the other roots; the 2πk term is what enumerates the full set.