1. 7.6 Powers and Roots of Complex Numbers
Powers of Complex numbers
In the same way,

2. 7.6 De Moivre’s Theorem De Moivre’s Theorem
If r(cos  + i sin ) is a complex number, and n is any real number, then
In compact form, this is written

3. 7.6 Finding a Power of a Complex Number
Example Find and express the result in
rectangular form.
Solution
Convert to trigonometric form.
480º and 120º are coterminal.
cos120º = -1/2;
sin120º =
Rectangular form

4. 7.6 Roots of Complex Numbers
To find three complex cube roots of
8(cos 135º + i sin 135º), for example, look for a complex number, say r(cos  + sin ), that will satisfy
nth Root
For a positive integer n, the complex number a+bi is the nth of the complex number x + yi if (a + bi)n = x + yi.

5. 7.6 Roots of Complex Numbers
By De Moivre’s Theorem,
becomes
Therefore, we must have r3 = 8, or r = 2, and

6. 7.6 Roots of Complex Numbers
Let k take on integer values 0, 1, and 2.
It can be shown that for integers k = 3, 4, and 5, these
values have repeating solutions. Therefore, all of the cube roots
(three of them) can be found by letting k = 0, 1, and 2.

7. 7.6 Roots of Complex Numbers
When k = 0, the root is 2(cos 45º + i sin 45º).
When k = 1, the root is 2(cos 165º + i sin 165º).
When k = 2, the root is 2(cos 285º + i sin 285º).
nth Root Theorem
If n is any positive integer, r is a positive real number, and  is in degrees, then the nonzero complex number r(cos  + i sin  ) has exactly n distinct nth roots, given by
where

8. 7.6 Finding Complex Roots
Example Find the two square roots of 4i. Write the
roots in rectangular form, and check your results
directly with a calculator.
Solution First write 4i in trigonometric form as
Here, r = 4 and  = /2. The square roots have modulus
and arguments as follows.

9. 7.6 Finding Complex Roots Since there are two roots, let k = 0 and 1.
If k = 0, then
If k = 1, then
Using these values for , the
square roots are 2 cis and
2 cis which can be written
in rectangular form as

10. 7.6 Finding Complex Roots
Example Find all fourth roots of Write the roots in
rectangular form.
Solution
If k = 0, then  = 30º + 90º·0 = 30º.
If k = 1, then  = 30º + 90º·1 = 120º.
If k = 2, then  = 30º + 90º·2 = 210º.
If k = 3, then  = 30º + 90º·3 = 300º.

11. 7.6 Finding Complex Roots
Using these angles on the previous slide, the fourth
roots are
2 cis 30º, 2 cis 120º, 2 cis 210º, and 2 cis 300º.
These four roots can be written in rectangular form as

12. 7.6 Solving an Equation by Finding Complex Roots
Example Find all complex number solutions of
x5 – 1 = 0. Graph them as vectors in the complex plane.
Solution Write the equation as
To find the five complex number solutions, write 1 in
polar form as
The modulus of the fifth roots is

13. 7.6 Solving an Equation by Finding Complex Roots
The arguments are given by
Using these arguments, the fifth roots are
1(cos 0º + i sin 0º), k = 0
1(cos 72º + i sin 72º), k = 1
1(cos 144º + i sin 144º), k = 2
1(cos 216º + i sin 216º), k = 3
1(cos 288º + i sin 288º), k = 4