4.10: Powers and Roots of Complex Numbers
Manually calculating (simplifying) a statement such as:
Fortunately you will learn in this lesson that there is an alternative: De Moivre's Theorem. De Moivre's Theorem is really the only practical method for finding the powers or roots of a complex number, but there is a catch...
What must be done to a complex number before De Moivre's Theorem can be utilized?
Watch This
This lesson reviews the two opposite operations involving De Moivre's Theorem: finding powers, and finding roots. The video lessons review each operation separately.
Embedded Video:
a. Raising Complex Numbers to a Power: 
b. Finding Roots of Complex Numbers:  James Sousa: Determining the Nth Roots of a Complex Number 

Guidance
Powers of Complex Numbers
How do we raise a complex number to a power? Let’s start with an example
(4  4i)^{3} = (4  4i) x (4  4i) x (4  4i) =
In rectangular form, this can get very complex. What about in r cis θ form?
So the problem becomes
and using our multiplication rule from the previous section,
Notice, (a + bi)^{3} = r^{3} cis 3 θ
In words: Raise the rvalue to the same degree as the complex number is raised and then multiply that by cis of the angle multiplied by the number of the degree.
Reflecting on the example above, we can identify De Moivre's Theorem:
Let z = r(cos θ + i sin θ) be a complex number in rcisθ form. If n is a positive integer, z^{n} is


It should be clear that the polar form provides a much faster result for raising a complex number to a power than doing the problem in rectangular form.
Roots of Complex Numbers
I imagine you noticed long ago that when an new operation is presented in mathematics, the inverse operation often follows. That is generally because the inverse operation is often procedurally similar, and it makes good sense to learn both at the same time.
This is no exception:
 The inverse operation of finding a power for a number is to find a root of the same number.

 a) Recall from Algebra that any root can be written as x^{1/}^{n}

 b) Given that the formula for De Moivre’s theorem also works for fractional powers, the same formula can be used for finding roots:



z1/n=(a+bi)1/n=r1/ncis(θn)


Example A
Find the value of
and θ is in the 1^{st} quadrant, so
Using our equation from above:
Expanding cis form:
Finally we have
z^{4} = 8  13.856i
Example B
Find
Solution
First, rewriting in exponential form: (1 + i)^{½}
And now in polar form:
Expanding cis form,
Using the formula:
In decimal form, we get
=1.189( 0.924 + 0.383i)
=1.099 + 0.455i
To check, we will multiply the result by itself in rectangular form:
Example C
Find the value of
Solution
First we put
Use
let
Use De Moivre’s Equation to find the first solution:
Leave answer in cis form to find the remaining solutions:
n = 3 which means that the 3 solutions are
Note: It is not necessary to add
The three solutions are:
Each of these solutions, when graphed will be
Check any one of these solutions to see if the results are confirmed.
Checking the second solution:
Does (0.965 – 0.810i)^{3} or (0.965 – 0.810i) (0.965 – 0.810i) (0.965 – 0.810i)
Concept question followup
A complex number operation written in rectangular form, such as: 

Vocabulary
De Moivre's Theorem is the only practical manual method of identifying the powers or roots of complex numbers.
Guided Practice
1) What are the two square roots of i?
2) Calculate
3) Calculate \begin{align*}(\sqrt3 + i)^7\end{align*}
Solutions
1) Let \begin{align*}z = \sqrt{0 + i}\end{align*}
 \begin{align*}r = 1, \ \theta = \pi/2\end{align*} or \begin{align*}z = \left [1 \times \ cis \frac{\pi}{2} \right ]^{1/2}\end{align*} Utilizing De Moivre’s Theorem:
 \begin{align*}z_1 = \left [1 \times \ cis \frac{\pi}{4} \right ]\end{align*} or \begin{align*}z_2 = \left [1 \times \ cis \frac{5\pi}{4} \right ]\end{align*}
 \begin{align*}z_1 = 1\left ( \mbox{cos}\frac{\pi}{4} + i \ \mbox{sin}\frac{\pi}{4} \right )\end{align*} or \begin{align*}z_2 = 1 \left (\mbox{cos} \frac{5\pi}{4} + i \ \mbox{sin} \frac{5\pi}{4} \right )\end{align*}
 \begin{align*}z_1 = 0.707 + 0.707i\end{align*} or \begin{align*}z_2 = 0.707  0.707i\end{align*}
Check for z_{1} solution: (0.707 + 0.707i)^{2} = i?
0.500 + 0.500i + 0.500i + 0.500i^{2} = 0.500 + i + 0.500(1) or i
2) Let z = 1 or z = 1 + 0i Then the problem becomes find z^{1/4} = (1 + 0i)^{1/4}
 Since \begin{align*}r = 1 \ \theta = 0, \ z^{1/4} = [1 \times cis \ 0]^{1/4}\end{align*} with \begin{align*}z_1 = 1^{1/4} \left (\mbox{cos}\ \frac{0}{4} + i \ \mbox{sin}\ \frac{0}{4} \right )\end{align*} or \begin{align*}1(1 + 0)\end{align*} or \begin{align*}1\end{align*}
That root is not a surprise. Now use De Moivre’s to find the other roots:
 \begin{align*}z_2 = 1^{1/4} \left [\mbox{cos}\left (0 + \frac{\pi}{2}\right ) + i \ \mbox{sin}\left (0 + \frac{\pi}{2} \right ) \right ]\end{align*} Since there are 4 roots, dividing 2π by 4 yields 0.5π
 or 0 + i or just i \begin{align*}z_3 = 1^{1/4} \left [\mbox{cos}\left (0 + \frac{2\pi}{2}\right ) + i \ \mbox{sin}\left (0 + \frac{2\pi}{2} \right ) \right ]\end{align*} which yields z_{3} = 1
 Finally \begin{align*}z_4 = 1^{1/4} \left [\mbox{cos}\left (0 + \frac{3\pi}{2}\right ) + i \ \mbox{sin}\left (0 + \frac{3\pi}{2} \right ) \right ]\end{align*} or \begin{align*}z_4 = i\end{align*}
The four fourth roots of 1 are 1, i, 1 and i
3) To calculate \begin{align*}(\sqrt3 + i)^7\end{align*} start by converting to \begin{align*}rcis\end{align*} form First find \begin{align*}r\end{align*} Recall \begin{align*}r = \sqrt{\sqrt{3}^2 + 1^2}\end{align*}
 \begin{align*}r = \sqrt{3 + 1}\end{align*}
 \begin{align*}r = 2\end{align*}
If \begin{align*}cos \theta = \frac{\sqrt3}{2}\end{align*} and \begin{align*}sin \theta = \frac{1}{2}\end{align*} then \begin{align*}\theta = 30^o\end{align*} and is in quadrant I. Now that we have trigonometric form, the rest is easy:
 \begin{align*}(\sqrt{3} + i)^7 = [2 (cos 30^o + i sin 30^o)]^7\end{align*} ..... Write the original problem in \begin{align*}r cis\end{align*} form
 \begin{align*}2^7[ (cos (7 \cdot 30^o) + i sin(7 \cdot 30^o)]\end{align*} ..... De Moivre's Theorem
 \begin{align*}128 [\frac{\sqrt3}{2} + \frac{1}{2}i]\end{align*} ..... Simplify
 \begin{align*}(\sqrt{3} + i)^7 = 64\sqrt3  64i\end{align*} ..... Simplify again
\begin{align*}\therefore (\sqrt{3} + i)^7 = 64\sqrt3  64i\end{align*}
Practice
Perform indicated operation on these complex numbers:
 Divide: \begin{align*}\frac{2 + 3i}{1  i}\end{align*}
 Multiply: \begin{align*}(6  i)(6 + i)\end{align*}
 Multiply: \begin{align*}\left (\frac{\sqrt{3}}{2}  \frac{1}{2} i \right )^2\end{align*}
 Find the product using polar form: \begin{align*}(2 + 2i)(\sqrt{3}  i)\end{align*}
 Multiply: \begin{align*}2(\mbox{cos} \ 40^\circ + i \ \mbox{sin} \ 40^\circ) \bullet 4(\mbox{cos} \ 20^\circ + i \ \mbox{sin} \ 20^\circ)\end{align*}
 Multiply: \begin{align*}2 \left (\mbox{cos} \ \frac{\pi}{8} + i \ \mbox{sin} \ \frac{\pi}{8} \right ) \bullet 2 \left (\mbox{cos} \ \frac{\pi}{10} + i \ \mbox{sin} \ \frac{\pi}{10} \right )\end{align*}
 Divide: \begin{align*}2(\mbox{cos} \ 80^\circ + i \ \mbox{sin} \ 80^\circ) \div 6(\mbox{cos} \ 200^\circ + i \ \mbox{sin} \ 200^\circ)\end{align*}
 Divide: \begin{align*}3\ \mbox{cis}(130^\circ) \div 4\ \mbox{cis}(270^\circ)\end{align*}
Use De Moivre’s Theorem:
 \begin{align*}[3(\mbox{cos} \ 80^\circ + i \ \mbox{sin} \ 80^\circ)]^3\end{align*}
 \begin{align*}\left [\sqrt{2} \left (\mbox{cos}\ \frac{5\pi}{16} + i \ \mbox{sin} \ \frac{5\pi}{16} \right ) \right ]^4\end{align*}
 \begin{align*}\left (\sqrt{3}  i \right )^6\end{align*}
 Identify the 3 complex cube roots of \begin{align*}1 + i\end{align*}
 Identify the 4 complex fourth roots of \begin{align*}16i\end{align*}
 Identify the five complex fifth roots of \begin{align*}i\end{align*}
complex number
A complex number is the sum of a real number and an imaginary number, written in the form .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 .Image Attributions
Description
Learning Objectives
Here you will explore "De Moivre's Theorem," which will allow you to calculate powers and roots of complex numbers.