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Powers and Roots of Complex Numbers

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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 $z_1 = 7 \left( \frac{\pi}{2} \right)$ and $z_2 = 9 \left(\frac{\pi}{3} \right)$ find:

1. $z_1 z_2$
2. $\left( \frac{z_1}{z_2} \right)$
3. $\left( \frac{z_2}{z_1} \right)$
4. $(z_1)^2$
5. $(z_2)^3$

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Find the quotients

1. $2(cos 80^o + i sin 80^o) \div 6(cos 200^o + i sin 200^o)$
2. $3cis(130^o) \div 4cis(270^o)$
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Powers and Roots

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

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

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Use De Moivre’s Theorem:

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