What is the value of `((1=-+isqrt3)/(2))^(3n)+((-1+isqrt3)/(2))^(3n) where i=sqrt(-1)`?

Options

A. 1
B. `-1`
C. `i`
D. `-i`

Answer

B

Explanation

`((-1+isqrt3)/(2))^(n)+((-1isqrt3)/(2))^(n)`
we know that cube roots of a unit use `1, omega, omega^(2)`
`omega=(-1+isqrt3)/(2),omega^(2)=(-1-isqrt3)/(2)`
So, `((-1+isqrt3)/(2))^(n)+((-1-isqrt3)/(2))^(n)=omega^(n)+omega^(2n)`
Given, n is not multiple of 3. So, `n=1,2,4,5...` In any case, ` omega^(n)+omega^(2n)=omega+omega^(2)=-1`