Given `alpha,beta,` respectively, the fifth and the fourth non-real roots of units, then find the value of `(1+alpha)(1+beta)(1+alpha^2)(1+beta^2)(1+alpha^4)(1+beta^4)`

Answer

0

Explanation

As ` alpha` is the fifth nonreal root of unity, we have `alpha^(4) + alpha^(3) + alpha^(2) + alpha + 1=0`
`beta ` is the fouth nonreal root of unity . Therefore,
`beta^(3) + beta^(2) + beta + 1=0`
Now, `( 1 + alpha )(1 + alpha^(2))(1 + alpha^(4))(1+ beta)(1+ beta^(2))(1 + beta^(3))`
`= (1+ alpha + alpha^(2)+ alpha^(3) ) (1+alpha^(4)) (1+ beta + beta^(2) + beta^(3))(1+ beta^(3)) (because 1+ beta + beta^(2) + beta^(3) = 0)`
`= 0`