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

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

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

Example 5b: Using De Moivre’s Theorem Find 1+𝑖 3 3 using De Moivre’s Theorem. Write your answer in standard 𝑎+𝑏𝑖 form. Check your answer using a graphing calculator.

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

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

Calculate the 5th roots of 32 𝑐𝑖𝑠 60°.

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