Question

I know that these answers are correct, but I don't understand how to get it using the De Moivere Theorem. Can someone please explain in detail? Thank you so much!

Answer 1

Explanation:

10.

First, convert 1+i from Cartesian to polar.

r = √(1² + 1²)

r = √2

θ = atan(1/1), θ in first quadrant

θ = 45°

Therefore:

(1+i)²⁰ = (√2 (cos 45° + i sin 45°))²⁰

(1+i)²⁰ = 1024 (cos 45° + i sin 45°)²⁰

Now applying the De Moivre theorem:

1024 (cos (20×45°) + i sin (20×45°))

1024 (cos (900°) + i sin (900°))

1024 (-1 + 0)

-1024

11.

Repeat the same steps from Question 10. First, convert to polar:

r = √(1² + (-1)²)

r = √2

θ = atan(-1/1), θ in fourth quadrant

θ = 315°

Therefore:

(1−i)¹⁰ = (√2 (cos 315° + i sin 315°))¹⁰

(1−i)¹⁰ = 32 (cos 315° + i sin 315°)¹⁰

Now applying the De Moivre theorem:

32 (cos (10×315°) + i sin (10×315°))

32 (cos (3150°) + i sin (3150°))

32 (0 − i)

-32i