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# DeMoivre's Theorem and nth Roots

## Raise complex numbers to powers or find their roots.

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Practice DeMoivre's Theorem and nth Roots
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Polar Theorems

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### Theorem Overview

##### In your own words, describe each theorem as it relates to polar equations.
 Theorem Description Product Theorem ________________________________________________________________ Quotient Theorem ________________________________________________________________ DeMoivre's Theorem ________________________________________________________________

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##### Complete the theorems:

$r_1(\cos \theta_1 + i \sin \theta_1) \cdot r_2(\cos \theta_2 + i \sin \theta_2)=$ ____________________________________

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$\frac{r_1(\cos \theta_1+i \sin \theta_1)}{r_2(\cos \theta_2+i \sin \theta_2)}=$ ____________________________________

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$z^n = [r(\cos \theta + i \sin \theta)]^n =$ ____________________________________

Where $z = r(\cos \theta + i \sin \theta)$ and let $n$ be a positive integer.

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The general rule for finding the $n^{th}$ roots of a complex number if $z = r(\cos \theta + i \sin \theta)$ is _____________________________________________ (Hint: start wth DeMoivre's Theorem).

### Practice

Multiply each pair of complex numbers. If they are not in trigonometric form, change them before multiplying.

1. $-3(\cos 70^\circ+i\sin 70^\circ)\cdot 3(\cos 85^\circ +i\sin 85^\circ )$
2. $7(\cos 85^\circ+i\sin 85^\circ)\cdot \sqrt{2}(\cos 40^\circ +i\sin 40^\circ )$
3. $(3-2i)\cdot (1+i)$

Divide each pair of complex numbers. If they are not in trigonometric form, change them before dividing.
1. $\frac{-3(\cos 70^\circ+i\sin 70^\circ)}{3(\cos 85^\circ +i\sin 85^\circ )}$
2. $\frac{7(\cos 85^\circ+i\sin 85^\circ)}{\sqrt{2}(\cos 40^\circ +i\sin 40^\circ )}$
3. $\frac{(3-2i)}{(1+i)}$
Use DeMoivre's Theorem to evaluate each expression. Write your answer in standard form.
1. $[3(\cos\frac{\pi}{4}+i\sin\frac{\pi}{4})]^5$
2. $(2-\sqrt{5}i)^5$
3. $(\sqrt{2}+\sqrt{2}i)^4$

Find the principal fifth roots of each complex number. Write your answers in standard form.

1. $32(\cos \frac{\pi}{4}+i\sin \frac{\pi}{4})$
2. $2(\cos \frac{\pi}{3}+i\sin \frac{\pi}{3})$
3. $32i$
Solve each equation.
1. $x^3=343$
2. $x^7=-128$
3. $x^4+5=86$