Question

Which of the following expressions is a sixth root of unity?

-1
i
-i
all of the above

Answer 1

-1

Explanation:

Answer 2

Let sixth root of unity be x

So
`x^(6)=1`

`x^6-1=0`

`(x^3)^2-1^2=0`

`(x^3-1)(x^3+1)=0`
`x^3-1=0 $ or $ x^3+1=0`

`(x-1)(x^2+x+1)=0 $ or $ (x+1)(x^2-x+1)=0`
`x-1=0 $ or $ x^2+x+1=0 $ or $ x+1=0 $ or $ x^2-x+1=0`

Using Quadratic formula for the quadratic equations we get

`x=1 $ or $ x=(-1\pm√(1-4) )/(2) $ or $ x = -1 $ or $ x=(1\pm√(1-4) )/(2)`

`x = 1 $ or $ x=(-1\pm√(3)i )/(2) $ or $ x = -1 $ or $ x=(1\pm√(3)i )/(2)`

Only 1, out of the given option is the sixth root of unity.