Test Objectives
• Demonstrate the ability to find powers of complex numbers
• Demonstrate the ability to find roots of complex numbers
• Demonstrate the ability to solve equations using complex roots
• Demonstrate the ability to graph complex roots on the Argand diagram
De Moivres Theorem Practice Test:

#1:

Instructions: Simplify, write your answer in rectangular form.

$$a)\hspace{.1em}[5(\text{cos}\hspace{.1em}330° + i \hspace{.1em}\text{sin}\hspace{.1em}330°)]^{4}$$

$$b)\hspace{.1em}[2(\text{cos}\hspace{.1em}60° + i \hspace{.1em}\text{sin}\hspace{.1em}60°)]^{2}$$

#2:

Instructions: Simplify, write your answer in polar form.

$$a)\hspace{.1em}\left(-\frac{5\sqrt{2}}{2}+ \frac{5\sqrt{2}}{2}i\right)^{3}$$

$$b)\hspace{.1em}\left(-\frac{3\sqrt{2}}{2}- \frac{3\sqrt{2}}{2}i\right)^{5}$$

#3:

Instructions: Find all nth roots. Write the answer in polar form.

$$a)\hspace{.1em}\frac{3}{2}- \frac{3\sqrt{3}}{2}i, n=3$$

$$b)\hspace{.1em}{-}3 + 3i\sqrt{3}, n=3$$

#4:

Instructions: Find all nth roots. Write the answer in polar form.

$$a)\hspace{.1em}\frac{5\sqrt{2}}{2}+ \frac{5\sqrt{2}}{2}i, n=3$$

$$b)\hspace{.1em}{-}2\sqrt{3}+ 2i, n=4$$

#5:

Instructions: Find and graph all complex solutions in polar form.

$$a)\hspace{.1em}x^{4}- i=0$$

$$b)\hspace{.1em}x^{3}- (4 + 4i\sqrt{3})=0$$

Written Solutions:

#1:

Solutions:

$$a)\hspace{.1em}{-}\frac{625}{2}- \frac{625\sqrt{3}}{2}i$$

$$b)\hspace{.1em}{-}2 + 2i\sqrt{3}$$

#2:

Solutions:

$$a)\hspace{.1em}125\left(\text{cos}\hspace{.1em}\frac{π}{4}+ i \hspace{.1em}\text{sin}\frac{π}{4}\right)$$

$$b)\hspace{.1em}243(\text{cos}\hspace{.1em}45° + i \hspace{.1em}\text{sin}\hspace{.1em}45°)$$

#3:

Solutions:

$$a)\hspace{.1em}\sqrt[3]{3}(\text{cos}\hspace{.1em}100° + i \hspace{.1em}\text{sin}\hspace{.1em}100°)$$ $$\sqrt[3]{3}(\text{cos}\hspace{.1em}220° + i \hspace{.1em}\text{sin}\hspace{.1em}220°)$$ $$\sqrt[3]{3}(\text{cos}\hspace{.1em}340° + i \hspace{.1em}\text{sin}\hspace{.1em}340°)$$

$$b)\hspace{.1em}\sqrt[3]{6}(\text{cos}\hspace{.1em}40° + i \hspace{.1em}\text{sin}\hspace{.1em}40°)$$ $$\sqrt[3]{6}(\text{cos}\hspace{.1em}160° + i \hspace{.1em}\text{sin}\hspace{.1em}160°)$$ $$\sqrt[3]{6}(\text{cos}\hspace{.1em}280° + i \hspace{.1em}\text{sin}\hspace{.1em}280°)$$

#4:

Solutions:

$$a)\hspace{.1em}\sqrt[3]{5}\left(\text{cos}\hspace{.1em}\frac{π}{12}+ i \hspace{.1em}\text{sin}\frac{π}{12}\right)$$ $$\sqrt[3]{5}\left(\text{cos}\hspace{.1em}\frac{3π}{4}+ i \hspace{.1em}\text{sin}\frac{3π}{4}\right)$$ $$\sqrt[3]{5}\left(\text{cos}\hspace{.1em}\frac{17π}{12}+ i \hspace{.1em}\text{sin}\frac{17π}{12}\right)$$

$$b)\hspace{.1em}\sqrt[4]{4}\left(\text{cos}\hspace{.1em}\frac{5π}{24}+ i \hspace{.1em}\text{sin}\frac{5π}{24}\right)$$ $$\sqrt[4]{4}\left(\text{cos}\hspace{.1em}\frac{17π}{24}+ i \hspace{.1em}\text{sin}\frac{17π}{24}\right)$$ $$\sqrt[4]{4}\left(\text{cos}\hspace{.1em}\frac{29π}{24}+ i \hspace{.1em}\text{sin}\frac{29π}{24}\right)$$ $$\sqrt[4]{4}\left(\text{cos}\hspace{.1em}\frac{41π}{24}+ i \hspace{.1em}\text{sin}\frac{41π}{24}\right)$$

#5:

Solutions:

$$a)\hspace{.1em}\text{cos}\hspace{.1em}22.5° + i \hspace{.1em}\text{sin}\hspace{.1em}22.5°$$ $$\text{cos}\hspace{.1em}112.5° + i \hspace{.1em}\text{sin}\hspace{.1em}112.5°$$ $$\text{cos}\hspace{.1em}202.5° + i \hspace{.1em}\text{sin}\hspace{.1em}202.5°$$ $$\text{cos}\hspace{.1em}292.5° + i \hspace{.1em}\text{sin}\hspace{.1em}292.5°$$

$$b)\hspace{.1em}2(\text{cos}\hspace{.1em}20° + i \hspace{.1em}\text{sin}\hspace{.1em}20°)$$ $$2(\text{cos}\hspace{.1em}140° + i \hspace{.1em}\text{sin}\hspace{.1em}140°)$$ $$2(\text{cos}\hspace{.1em}260° + i \hspace{.1em}\text{sin}\hspace{.1em}260°)$$