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Powers and Roots of Complex Numbers

De Moivre's Theorem using polar form.

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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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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 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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Click here for answers.

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