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# Powers and Roots of Complex Numbers

## De Moivre's Theorem using polar form.

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Practice Powers and Roots of Complex Numbers
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Estimated10 minsto complete
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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 z1=7(π2)\begin{align*}z_1 = 7 \left( \frac{\pi}{2} \right)\end{align*} and z2=9(π3)\begin{align*} z_2 = 9 \left(\frac{\pi}{3} \right)\end{align*} find:

1. z1z2\begin{align*}z_1 z_2\end{align*}
2. (z1z2)\begin{align*}\left( \frac{z_1}{z_2} \right)\end{align*}
3. (z2z1)\begin{align*}\left( \frac{z_2}{z_1} \right)\end{align*}
4. (z1)2\begin{align*}(z_1)^2\end{align*}
5. (z2)3\begin{align*}(z_2)^3\end{align*}

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Find the quotients

1. 2(cos80o+isin80o)÷6(cos200o+isin200o)\begin{align*} 2(cos 80^o + i sin 80^o) \div 6(cos 200^o + i sin 200^o)\end{align*}
2. 3cis(130o)÷4cis(270o)\begin{align*} 3cis(130^o) \div 4cis(270^o)\end{align*}
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Click here for help with the Product and Quotient theorems.

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##### Powers and Roots

To find roots of complex numbers, you De Moivre's Theorem to creat the equation z1/n=(a+bi)1/n=r1/ncis(θn)\begin{align*}z^{1/n} = (a + bi)^{1/n} = r^{1/n} cis \left ( \frac{\theta}{n} \right )\end{align*}. 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(cos 80+i sin 80)]3\begin{align*}[3(\mbox{cos} \ 80^\circ + i \ \mbox{sin} \ 80^\circ)]^3\end{align*}
2. [2(cos 5π16+i sin 5π16)]4\begin{align*}\left [\sqrt{2} \left (\mbox{cos}\ \frac{5\pi}{16} + i \ \mbox{sin} \ \frac{5\pi}{16} \right ) \right ]^4\end{align*}
3. (3i)6\begin{align*}\left (\sqrt{3} - i \right )^6\end{align*}
4. Identify the 3 complex cube roots of 1+i\begin{align*}1 + i\end{align*}
5. Identify the 4 complex fourth roots of 16i\begin{align*}-16i\end{align*}
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