Question

How do you use DeMoivre's theorem to simplify `(1+i)^4`?

Answer 1

`(1+i)^4 = -4`

Explanation:

Note that

`1+i = sqrt(2)(cos (pi/4) + i sin (pi/4))`

De Moivre tells us that:

`(cos theta + i sin theta)^n = cos n theta + i sin n theta`

So we find:

`(1+i)^4 = (sqrt(2)(cos (pi/4) + i sin (pi/4)))^4`

`color(white)((1+i)^4) = (sqrt(2))^4(cos (pi/4) + i sin (pi/4))^4`

`color(white)((1+i)^4) = 4(cos pi + i sin pi)`

`color(white)((1+i)^4) = 4((-1) + i (0))`

`color(white)((1+i)^4) = -4`