- 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
#1:
Instructions: Simplify, write your answer in rectangular form.
$$a)\hspace{.1em}[5(\text{cos}\hspace{.1em}330° + i \hspace{.1em}\text{sin}\hspace{.1em}330°)]^{4}$$
$$b)\hspace{.1em}[2(\text{cos}\hspace{.1em}60° + i \hspace{.1em}\text{sin}\hspace{.1em}60°)]^{2}$$
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#2:
Instructions: Simplify, write your answer in polar form.
$$a)\hspace{.1em}\left(-\frac{5\sqrt{2}}{2}+ \frac{5\sqrt{2}}{2}i\right)^{3}$$
$$b)\hspace{.1em}\left(-\frac{3\sqrt{2}}{2}- \frac{3\sqrt{2}}{2}i\right)^{5}$$
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#3:
Instructions: Find all nth roots. Write the answer in polar form.
$$a)\hspace{.1em}\frac{3}{2}- \frac{3\sqrt{3}}{2}i, n=3$$
$$b)\hspace{.1em}{-}3 + 3i\sqrt{3}, n=3$$
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#4:
Instructions: Find all nth roots. Write the answer in polar form.
$$a)\hspace{.1em}\frac{5\sqrt{2}}{2}+ \frac{5\sqrt{2}}{2}i, n=3$$
$$b)\hspace{.1em}{-}2\sqrt{3}+ 2i, n=4$$
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#5:
Instructions: Find and graph all complex solutions in polar form.
$$a)\hspace{.1em}x^{4}- i=0$$
$$b)\hspace{.1em}x^{3}- (4 + 4i\sqrt{3})=0$$
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Written Solutions:
#1:
Solutions:
$$a)\hspace{.1em}{-}\frac{625}{2}- \frac{625\sqrt{3}}{2}i$$
$$b)\hspace{.1em}{-}2 + 2i\sqrt{3}$$
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#2:
Solutions:
$$a)\hspace{.1em}125\left(\text{cos}\hspace{.1em}\frac{π}{4}+ i \hspace{.1em}\text{sin}\frac{π}{4}\right)$$
$$b)\hspace{.1em}243(\text{cos}\hspace{.1em}45° + i \hspace{.1em}\text{sin}\hspace{.1em}45°)$$
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#3:
Solutions:
$$a)\hspace{.1em}\sqrt[3]{3}(\text{cos}\hspace{.1em}100° + i \hspace{.1em}\text{sin}\hspace{.1em}100°)$$ $$\sqrt[3]{3}(\text{cos}\hspace{.1em}220° + i \hspace{.1em}\text{sin}\hspace{.1em}220°)$$ $$\sqrt[3]{3}(\text{cos}\hspace{.1em}340° + i \hspace{.1em}\text{sin}\hspace{.1em}340°)$$
$$b)\hspace{.1em}\sqrt[3]{6}(\text{cos}\hspace{.1em}40° + i \hspace{.1em}\text{sin}\hspace{.1em}40°)$$ $$\sqrt[3]{6}(\text{cos}\hspace{.1em}160° + i \hspace{.1em}\text{sin}\hspace{.1em}160°)$$ $$\sqrt[3]{6}(\text{cos}\hspace{.1em}280° + i \hspace{.1em}\text{sin}\hspace{.1em}280°)$$
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#4:
Solutions:
$$a)\hspace{.1em}\sqrt[3]{5}\left(\text{cos}\hspace{.1em}\frac{π}{12}+ i \hspace{.1em}\text{sin}\frac{π}{12}\right)$$ $$\sqrt[3]{5}\left(\text{cos}\hspace{.1em}\frac{3π}{4}+ i \hspace{.1em}\text{sin}\frac{3π}{4}\right)$$ $$\sqrt[3]{5}\left(\text{cos}\hspace{.1em}\frac{17π}{12}+ i \hspace{.1em}\text{sin}\frac{17π}{12}\right)$$
$$b)\hspace{.1em}\sqrt[4]{4}\left(\text{cos}\hspace{.1em}\frac{5π}{24}+ i \hspace{.1em}\text{sin}\frac{5π}{24}\right)$$ $$\sqrt[4]{4}\left(\text{cos}\hspace{.1em}\frac{17π}{24}+ i \hspace{.1em}\text{sin}\frac{17π}{24}\right)$$ $$\sqrt[4]{4}\left(\text{cos}\hspace{.1em}\frac{29π}{24}+ i \hspace{.1em}\text{sin}\frac{29π}{24}\right)$$ $$\sqrt[4]{4}\left(\text{cos}\hspace{.1em}\frac{41π}{24}+ i \hspace{.1em}\text{sin}\frac{41π}{24}\right)$$
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#5:
Solutions:
$$a)\hspace{.1em}\text{cos}\hspace{.1em}22.5° + i \hspace{.1em}\text{sin}\hspace{.1em}22.5°$$ $$\text{cos}\hspace{.1em}112.5° + i \hspace{.1em}\text{sin}\hspace{.1em}112.5°$$ $$\text{cos}\hspace{.1em}202.5° + i \hspace{.1em}\text{sin}\hspace{.1em}202.5°$$ $$\text{cos}\hspace{.1em}292.5° + i \hspace{.1em}\text{sin}\hspace{.1em}292.5°$$ 
$$b)\hspace{.1em}2(\text{cos}\hspace{.1em}20° + i \hspace{.1em}\text{sin}\hspace{.1em}20°)$$ $$2(\text{cos}\hspace{.1em}140° + i \hspace{.1em}\text{sin}\hspace{.1em}140°)$$ $$2(\text{cos}\hspace{.1em}260° + i \hspace{.1em}\text{sin}\hspace{.1em}260°)$$ 
