Question

How do you find the fourth roots of `-4`?

Answer 1

`4^(1/4)(e^(pi/4i), e^(3pi/4i), e^(5pi/4i), e^(7pi/4i))`
`=+-1+-i`

Explanation:

Use, #cos (2n+1)pi = -1, n =0, 1, 3, 3, ... and De Moivre's theorem.

`(-4)^(1/4) = (4(-1)=4e^((2n+1)pi i))^(1/4), n=0, 1, 2, 3, ...`

`=4^(1/4)e^(((2n+1)/4)i),n=0, 1, 2, 3, ..`,

`=4^(1/4)(e^(pi/4i), e^(3pi/4i), e^(5pi/4i), e^(7pi/4i))`, repeated in a cycle

`=sqet 2/sqrt 2(+-1+-i)`

`=(+-1+-i)`