1. De Moivre’s Theorem and nth Roots
Goal: Find powers and roots of complex numbers.
6.6 Day 3
De Moivre’s Theorem and nth Roots

2. Powers of Complex Numbers
Given: 𝑧=𝑟 𝑐𝑖𝑠𝜃
𝑧 𝑛 =
(DeMoivre’s Theorem)

3. Example 5a: Using De Moivre’s Theorem
Calculate 2 𝑐𝑖𝑠 20° Write your answer in standard 𝑎+𝑏𝑖 form.

4. Example 5b: Using De Moivre’s Theorem
Find 1+𝑖 using De Moivre’s Theorem. Write your answer in standard 𝑎+𝑏𝑖 form. Check your answer using a graphing calculator.

5. Let 𝑧=2 𝑐𝑖𝑠 60°. Find 𝑧 𝑛 for n = 0, 1, 2, 3, 4.

6. Roots of Complex Numbers
Given: 𝑧=𝑟 𝑐𝑖𝑠𝜃
𝑛 𝑧 =

7. Calculate the 5th roots of 32 𝑐𝑖𝑠 60°.

8. Example 8: Finding Cube Roots
Find the cube roots of -1. Write the roots in both trigonometric and standard form.