If `x=a+b,y=a gamma+b beta and z=abeta+ b gamma` , where `gamma and beta` are the imaginary cube roots ofunity, then `xyz=`

If x=a+b,y=a gamma+b beta and z=a beta+b gamma, where gamma and beta are the imaginary cube roots ofunity,then xyz=

If x=a+b,y=a alpha+b beta and z=abeta+ b alpha , where alpha and beta are the imaginary cube roots ofunity, then xyz=

If x=a+b,y=a alpha+b beta and z=abeta+ b alpha , where alpha and beta are the imaginary cube roots ofunity, then xyz=

If x=a+b, y=a alpha + b beta and z= alpha beta + b alpha , where alpha and beta are complex cube roots of unity, then xyz=

If alpha , beta , gamma in {1,omega,omega^(2)} (where omega and omega^(2) are imaginery cube roots of unity), then number of triplets (alpha,beta,gamma) such that |(a alpha+b beta+c gamma)/(a beta+b gamma+c alpha)|=1 is

If alpha , beta , gamma in {1,omega,omega^(2)} (where omega and omega^(2) are imaginery cube roots of unity), then number of triplets (alpha,beta,gamma) such that |(a alpha+b beta+c gamma)/(a beta+b gamma+c alpha)|=1 is

If p lt 0 and alpha, beta, gamma are cube roots of p then for any a,b and c show that (aalpha+b beta + cgamma)/(a beta+bgamma +calpha)=omega^(2) where omega is an imaginary cube root of 1.

If 1, omega, omega^(2) are the three cube roots of unity and alpha, beta, gamma are the cube roots of p, plt0 , then for any x, y, z the expression (x alpha+y beta+z gamma)/(x beta+y gamma+z alpha)=