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