<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
You are viewing an older version of this Study Guide. Go to the latest version.

# Powers and Roots of Complex Numbers

## De Moivre's Theorem using polar form.

Estimated17 minsto complete
%
Progress
Practice Powers and Roots of Complex Numbers

MEMORY METER
This indicates how strong in your memory this concept is
Progress
Estimated17 minsto complete
%
Complex Number Theorems

### Vocabulary

##### Fill in the equation for each theorem then describe each theorem in your own words.
 Theorem Equation Description Product Theorem __________________________ _________________________________________________ Quotient Theorem __________________________ _________________________________________________ De Moivre's Theorem __________________________ _________________________________________________

### Practice

##### Product and Quotient Theorems

If \begin{align*}z_1 = 7 \left( \frac{\pi}{2} \right)\end{align*} and \begin{align*} z_2 = 9 \left(\frac{\pi}{3} \right)\end{align*} find:

1. \begin{align*}z_1 z_2\end{align*}
2. \begin{align*}\left( \frac{z_1}{z_2} \right)\end{align*}
3. \begin{align*}\left( \frac{z_2}{z_1} \right)\end{align*}
4. \begin{align*}(z_1)^2\end{align*}
5. \begin{align*}(z_2)^3\end{align*}

.

Find the quotients

1. \begin{align*} 2(cos 80^o + i sin 80^o) \div 6(cos 200^o + i sin 200^o)\end{align*}
2. \begin{align*} 3cis(130^o) \div 4cis(270^o)\end{align*}
.

.

##### Powers and Roots

To find roots of complex numbers, you De Moivre's Theorem to creat the equation \begin{align*}z^{1/n} = (a + bi)^{1/n} = r^{1/n} cis \left ( \frac{\theta}{n} \right )\end{align*}. This helps you find roots.

Remember: Numbers must be in polar form to use De Moivre's Theorem!

.

Use De Moivre’s Theorem:

1. \begin{align*}[3(\mbox{cos} \ 80^\circ + i \ \mbox{sin} \ 80^\circ)]^3\end{align*}
2. \begin{align*}\left [\sqrt{2} \left (\mbox{cos}\ \frac{5\pi}{16} + i \ \mbox{sin} \ \frac{5\pi}{16} \right ) \right ]^4\end{align*}
3. \begin{align*}\left (\sqrt{3} - i \right )^6\end{align*}
4. Identify the 3 complex cube roots of \begin{align*}1 + i\end{align*}
5. Identify the 4 complex fourth roots of \begin{align*}-16i\end{align*}
.