11-3

11-3 Powers of Complex Numbers
This week I will be reviewing how to find powers of complex numbers. We use De Moivre's Theorem to find these roots. It's simpler to find powers of a complex number when the complex number is in polar form.
De Moivre's Theorem:
If z = r cis theta, then z^n = r^n cis n(theta).

*In this section, you will not draw Argand diagrams.*

Example 1: Evaluate (2 cis 45 degrees)^2.

• z^2 = 2^2 cis 2(45 degrees)


• z^2 = 4 cis 90 degrees

Example 2: z = 3 cis 10 degrees. Use De Moivre's Theorem to find z^3.

• z^3 = 3^3 cis 3(10 degrees)


• z^3 = 27 cis 30 degrees

Example 3: If z = 1 + c, find z^4.

• z = x + yi


• r = sq. root of x^2 + y^2


• r = +/- sq. root of 2


• theta = tan^-1(y/x)


• theta = 45 degrees, 225 degrees


• z = sq. root of 2 cis 45 degrees


• z = sq. root of 2 cos 45 degrees + sq. root of 2 sin 45 degrees(i)


• z = - sq. root of 2 cos 225 degrees


• z^4 = sq. root of 2^4 cos 4(45 degrees)


• = 4 cis 180 degrees


• = r cos 180 degrees + 4 sin 180i