Question

How do you use DeMoivre's Theorem to find `(1+i)^20` in standard form?

Answer 1

Divide by the norm of `1+i` and apply De Moivre's theorem to find that

`(1+i)^20=-1024`

Explanation:

De Moivre's formula, which can be derived from Euler's formula that `e^(itheta)=cos(theta)+isin(theta)`, states that

`(cos(theta)+isin(theta))^n = cos(ntheta)+isin(ntheta)`

However, as `1+i` does not lie on the unit circle, we cannot use the formula directly. To fix this, we can divide `1+i` by its norm `sqrt(2)` and factor out the necessary constant:

`(1+i)^20 = (sqrt(2)(1/sqrt(2)+1/sqrt(2)i))^20`

`=(sqrt(2))^20(sqrt(2)/2+sqrt(2)/2i)^20`

`=1024(cos(pi/4)+isin(pi/4))^20`

`=1024(cos(20*pi/4)+isin(20*pi/4))`

`=1024(cos(5pi)+isin(5pi))`

`=1024(-1+i*0)`

`=-1024`