Section 4.5 DeMoivre's Theorem

Section Objectives

  1. Compute whole number powers of complex numbers.
  2. Compute th roots of complex numbers.



Powers of Complex Numbers

We have already seen some of the advantages of writing complex numbers in polar form. Another very nice result is DeMoivre's theorem. It follows from our multiplication rule.


DeMoivre's Theorem

If is a complex number and is a positive integer, then

.




Example







Roots of Complex Numbers

DeMoivre's theorem can also be used "in reverse" to find th roots of complex numbers.


For the positive integer , the complex number has exactly distinct th roots, , given by

,

where .



Examples