If `omega` is an imaginary cube root of unity, then show that `(1-omega)(1-omega^2)(1-omega^4) (1-omega^5)=9`

Options

A. `2^(3n)`
B. `2^(2n)`
C. `2^(n)`
D. none of these

Answer

C

Explanation

We have,
`(1+omega)(1+omega^(2))(1+omega^(3))(1+omega^(4))(1+omega^(5))(1+omega^(6))….(1+omega^(3n))`
`=[{(1+omega)(1+omega^(2))}{(1+omega^(4))(1+omega^(5))}{(1+omega^(7))(1+omega^(8))}……{(1+omega)^(3n-2)(1+omega)^(3n-1)} xx (1 xx omega^(3))(1+omega^(6))(1+omega^(9))....(1+omega^(3n)] = 2^(n)[{(-omega^(2))(-omega)}{(-omega^(2))(-omega)}.....{(-omega^(2))(-omega)}]=2^(n)`