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