Section 10.5 (d) - DeMoivre's Theorem

Section Objectives

1. Compute whole number powers of complex numbers.
2. Compute $n$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 $z=r(\cos \theta + i \sin \theta)$ is a complex number and $n$ is a positive integer, then

$z^n = r^n(\cos n\theta + i \sin n\theta)$.

Example

• Use DeMoivre's theorem to compute $(-1+\sqrt{3} \, i)^{12}$.

• Use DeMoivre's theorem to compute $(-1- i)^4$.

Roots of Complex Numbers

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

For the positive integer $n$, the complex number $z=r(\cos \theta + i \sin \theta)$ has exactly $n$ distinct $n$th roots, $z_1, z_2, \dots, z_n$, given by

$z_k=\sqrt[n]{r} \displaystyle \left( \cos \frac{\theta + 2\pi k}{n} + i \sin \frac{\theta + 2\pi k}{n} \right)$,

where $k=0,1,2,\dots,n-1$.

Examples

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

• Compute the three cube roots of $z=-2+2i$.