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DeMoivre's Theorem

Main formulas:

• If the complex number z is written in polar form z = re^iθ , then we can find n-th powers as follows:
  

  zn = (reiθ )n = [r(cosθ + i sinθ )]n = rn(cosn θ + i sinn θ ) = rn ei  nθ

• Every nonzero complex number has exactly n n-th roots.
• We can find n-th roots as follows:
  

  n √   z   = z[1/(n)] = (reiθ )[1/(n)] = [r(cosθ + i sinθ )][1/(n)] = r[1/(n)] ( cos θ + 2kπ n + i sin θ + 2kπ n ) = r[1/(n)] ei  [(θ + 2kπ)/n]

  where k = 0,1,2,..., n− 1.