Topic: Complex, Parametric, and Polar Forms

Subtopic: Powers and Roots in Trig Form

**Overview**

From an intermediate algebra course we know how to expand and simplify a Complex number in the form (a+bi)^{n} via multiple FOILs or using Pascal's Triangle. In this lesson we convert the Complex number to trig form and use a formula to evaluate (r·cisθ)^{n} which is a great time saver!

We also learn something we never learned to do in intermediate algebra, that is to take roots of Complex numbers in trig form nth-root((r·cisθ)=(r·cisθ)^{1/n}. Finding powers and roots of Complex numbers in trigonometric form use one of two formulas called __DeMoivre's Theorems__.

**Objectives**

By the end of this topic you should know and be prepared to be tested on:

- 9.3.1 Algebraically find the power of a Complex number in trigonometric form using DeMoivre's Theorem of Powers,

**z**^{n}=(r·cis(θ))^{n}=r^{n}cis(nθ) - 9.3.2 Algebraically find the roots of a Complex number in trigonometric form using DeMoivre's Theorem of Roots,

**the n nth-roots of z are z**^{1/n}=(r^{1/n}cis((θ+2πk)/n)) where k=0,1,2,...,n-1 - 9.3.3 Sketch the roots of a Complex number in trigonometric form on the Complex plane
- 9.3.4 Use trigonometric form and DeMoivre's Theorem of Roots to solve equations that have Complex solutions

**Terminology**

Terms you should be able to define: DeMoivre's Theorem of Powers, DeMoivre's Theorem of Roots