# 6.13: DeMoivre's Theorem and nth Roots

**At Grade**Created by: CK-12

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**Practice**DeMoivre's Theorem and nth Roots

You are in math class one day when your teacher asks you to find . Are you able to find roots of complex numbers? By the end of this Concept, you'll be able to perform this calculation.

### Watch This

James Sousa: Determining the Nth Roots of a Complex Number

### Guidance

Other Concepts in this course have explored all of the basic operations of arithmetic as they apply to complex numbers in standard form and in polar form. The last discovery is that of taking roots of complex numbers in polar form. Using De Moivre’s Theorem we can develop another general rule – one for finding the root of a complex number written in polar form.

As before, let and let the root of be . So, in general, and .

From this derivation, we can conclude that or and . Therefore, for any integer , is an root of if and . Therefore, the general rule for finding the roots of a complex number if is: . Let’s begin with a simple example and we will leave in degrees.

#### Example A

Find the two square roots of .

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Solution:
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Express
in polar form.

To find the other root, add to .

#### Example B

Find the three cube roots of

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Solution:
**
Express
in polar form:

In standard form: .

#### Example C

Calculate

Using the for of DeMoivres Theorem for fractional powers, we get:

### Vocabulary

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DeMoivres Theorem:
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DeMoivres theorem
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relates a complex number raised to a power to a set of trigonometric functions.

### Guided Practice

1. Find .

2. Find the principal root of . Remember the principal root is the positive root i.e. so the principal root is +3.

3. Find the fourth roots of .

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Solutions:
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1.

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2.

In standard form and this is the principal root of .

3.

in polar form is:

### Concept Problem Solution

Finding the two square roots of involves first converting the number to polar form:

For the radius:

And the angle:

To find the other root, add to .

### Practice

Find the cube roots of each complex number. Write your answers in standard form.

Find the principal fifth roots of each complex number. Write your answers in standard form.

- Find the sixth roots of -64 and plot them on the complex plane.
- How many solutions could the equation have? Explain.
- Solve . Use your answer to #13 to help you.

roots of unity

The roots of unity are the roots of the number 1.complex number

A complex number is the sum of a real number and an imaginary number, written in the form .complex plane

The complex plane is the graphical representation of the set of all complex numbers.De Moivre's Theorem

De Moivre's theorem is the only practical manual method for identifying the powers or roots of complex numbers. The theorem states that if is a complex number in form and is a positive integer, then .trigonometric polar form

To write a complex number in trigonometric form means to write it in the form . is shorthand for this expression.### Image Attributions

## Description

## Learning Objectives

Here you'll learn to simplify complex numbers to some root using DeMoivre's Theorem.