Question

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

Answer 1

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

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

= `3 + 2"i" sqrty(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

Now put n = 6

⇒ `(sqrt(3) + "i")^6`

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

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

= `8[1^3 + ("i" sqrt(3))^3 + 3"i" sqrt(3) (1 + "i" sqrt(3))]`

= `8(1 - 3 sqrt(3)"i" + 3sqrt(3)"i" - 9)`

= 8(– 8)

= – 64

which is purely real

∴ n = 6