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

Use DeMoivre's theorem to compute .

Use DeMoivre's theorem to compute .

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

Find all sixth roots of 1. (These are usually called "roots of unity".)