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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:

\begin{align*}r_1(\cos \theta_1 + i \sin \theta_1) \cdot r_2(\cos \theta_2 + i \sin \theta_2)=\end{align*} ____________________________________

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\begin{align*}\frac{r_1(\cos \theta_1+i \sin \theta_1)}{r_2(\cos \theta_2+i \sin \theta_2)}=\end{align*} ____________________________________

.

\begin{align*}z^n = [r(\cos \theta + i \sin \theta)]^n =\end{align*} ____________________________________

Where \begin{align*}z = r(\cos \theta + i \sin \theta)\end{align*} and let \begin{align*}n\end{align*} be a positive integer.

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The general rule for finding the \begin{align*}n^{th}\end{align*} roots of a complex number if \begin{align*}z = r(\cos \theta + i \sin \theta)\end{align*} 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. \begin{align*}-3(\cos 70^\circ+i\sin 70^\circ)\cdot 3(\cos 85^\circ +i\sin 85^\circ )\end{align*}
2. \begin{align*}7(\cos 85^\circ+i\sin 85^\circ)\cdot \sqrt{2}(\cos 40^\circ +i\sin 40^\circ )\end{align*}
3. \begin{align*}(3-2i)\cdot (1+i)\end{align*}

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

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

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