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Powers and Roots of Complex Numbers

De Moivre's Theorem using polar form.

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Complex Number Theorems


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 __________________________ _________________________________________________


Product and Quotient Theorems

If \begin{align*}z_1 = 7 \left( \frac{\pi}{2} \right)\end{align*} and \begin{align*} z_2 = 9 \left(\frac{\pi}{3} \right)\end{align*} find:

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


Find the quotients

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


Powers and Roots 

To find roots of complex numbers, you De Moivre's Theorem to creat the equation \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!


Use De Moivre’s Theorem:

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

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