If `alpha` is a non-real fifth root of unity, then the value of `3^(|1+alpha+alpha^(2),alpha^(-2)-alpha^(-1)|`, is

Options

A. 9
B. 1
C. `11//3`
D. none of these

Answer

A

Explanation

We have,
`alpha=1^(1//5), alpha ne 1`
`rArr alpha^(5)=1 rArr alpha^(-2)=alpha^(3)` and `alpha^(-1)=alpha^(4)`
`therefore 1+alpha+alpha^(2)+alpha^(-2)-alpha^(-1)`
`=1+alpha+alpha^(2)+alpha^(3)-alpha^(4)`
`=(alpha^(4)-1)/(alpha-1)-alpha^(4) = (1/alpha-1)/(alpha-1)-1/alpha` `[therefore alpha^(5)=1 therefore alpha^(4)=1/alpha]`
`=-1/alpha-1/alpha=-2/alpha=-2alphap^(4)`
`rArr |1+alpha+alpha^(2)+alpha^(-2)-alpha^(-1)|=|-2alpha^(4)|=2|alpha|^(4)=2`
`rArr 3^(|1+alpha+alpha^(2)+alpha^(-2)-alpha^(-1)|=3^(2)=9`