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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 z1=7(π2) and z2=9(π3) find:

  1. z1z2
  2. (z1z2)
  3. (z2z1)
  4. (z1)2
  5. (z2)3


Find the quotients

  1. 2(cos80o+isin80o)÷6(cos200o+isin200o)
  2. 3cis(130o)÷4cis(270o)
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 z1/n=(a+bi)1/n=r1/ncis(θn). This helps you find roots.

Remember: Numbers must be in polar form to use De Moivre's Theorem!


Use De Moivre’s Theorem:

  1. [3(cos 80+i sin 80)]3
  2. [2(cos 5π16+i sin 5π16)]4
  3. (3i)6
  4. Identify the 3 complex cube roots of 1+i
  5. Identify the 4 complex fourth roots of 16i

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