Question

If `((1 + "i"sqrt3)/(1 - "i"sqrt3))^"n"` is an integer, then n is ______.

Options

• 1

• 2

• 3

• 4

Answer 1

If `((1 + "i"sqrt3)/(1 - "i"sqrt3))^"n"` is an integer, then n is 3.

Explanation:

`(1 + "i"sqrt3)/(1 - "i"sqrt3) = ((1 + "i"sqrt3)/(1 - "i"sqrt3))((1 + "i"sqrt3)/(1 + "i"sqrt3))`

`= (- 2 + "i"2sqrt3)/4`

`= (- 1 + "i"sqrt3)/2 = omega`

`therefore ((1 + "i"sqrt3)/(1 - "i"sqrt3))^"n" = omega^"n" = omega^3 = 1 => "n" = 3`