Question

Find the least value of the positive integer n for which `(sqrt(3) + "i")^"n"` purely imaginary

Answer 1

Given `(sqrt(3) + "i")^"n"`

= `(sqrt(3)^2 + 2"i" sqrt(3) + ("i")^2`

= `3 + 2"i" sqrt(3) - 1`

= `2 + 2"i" sqrt(3)`

= `2(1 + "i" sqrt(3))`

Put n = 3 or 4 or 5

Then real part is not possible

Put n = 3

⇒ `(sqrt(3) + "i")^3`

= `(sqrt(3))^3 + ("i")^3 + 3"i" sqrt(3)(sqrt(3) + "i")`

= `3sqrt(3) - "i" + 9"i" - 3sqrt(3)`

= 8i

Which is purely imaginary

∴ n = 3