If 1, `alpha_(1), alpha_(2), alpha_(3),…….,alpha_(s)` are ninth roots of unity (taken in counter -clockwise sequence in the Argard plane). Then find the value of `|(2-alpha_(1))(2-alpha_(3)),(2-alpha_(5))(2-alpha_(7))`|.

Explanation

According to the question,
`x^(9)-1=(x-1)(x-alpha_(1))(x-alpha_(2))(x-alpha_(8))`
`:." "2^(9)-1=(2-alpha_(1))(2-alpha_(2))...(2-alpha_(8))`
Since `alpha_(1)" and "alpha_(8)` are conjugate to each other, `2-alpha_(1)" and "2-alpha_(8)` are also conjugates to each other.
`:." "|2-alpha_(1)|=|2-alpha_(8)|`
Similarly, `|2-alpha_(2)|=|2-alpha_(7)|,|2-alpha_(3)|=|2-alpha_(6)|" and "|2-alpha_(4)|=|2-alpha_(5)|.`
`:." "|(2-alpha_(1))(2-alpha_(3))(2-alpha_(5))(2-alpha_(7))|^(2)=2^(9)-1`
`:." "|(2-alpha_(1))(2-alpha_(3))(2-alpha_(5))(2-alpha_(7))|=sqrt(2^(9)-1)=sqrt(511)`