Prove that `(p+q omega+r omega^(2))/(r+p omega+q omega^(2))=omega^(2)`

Prove that (1-omega-omega^(2))(1-omega+omega^(2))(1+omega-omega^(2))=8

Prove that omega(1+omega-omega^(2))=-2

If omega is cube roots of unity, prove that {[(1,omega,omega^2),(omega,omega^2,1),(omega^2,1,omega)]+[(omega,omega^2,1),(omega^2,1,omega),(omega,omega^2,1)]} [(1),(omega),(omega^2)]=[(0),(0),(0)]

If,omega,omega'2be the imaginary cube roots of unity,then prove that (2-omega)(2-omega^(2))(2-omega^(10))(2-omega^(11))=49

If omega is an imaginary cube root of unity,then the value of (p+q)^(3)+(p omega+q omega^(2))^(3)+(p omega^(2)+q omega)^(3) is: (a)3(p+q)(p+q omega)(p+q omega^(2))(b)3(p^(3)+q^(3))(c)3(p+q)-pq(p+q)(d)3(p^(3)+q^(3))pq(p+q)

If 1,omega,omega^(2) are the cube roots of unity,then the value of (a omega^(2)+b+c omega+d omega^(2))/(a+b omega+c omega^(2)+d) (a,b,c,d in R) is

If p=a+b omega+c omega^(2);q=b+c omega+a omega^(2) and r=c+a omega+b omega^(2) where a,b,c!=0 and omega is the complex cube root of unity,then

{[(1,omega,omega^(2)),(omega,omega^(2),1),(omega^(2),1,omega)] + [(omega,omega^(2),1),(omega^(2),1,omega),(omega,omega^(2),1)]} [(1),(omega),(omega^(2))]

If x=p+q,y=p omega+q omega^(2), and z=p omega^(2)+q omega where omega is a complex cube root of unity then xyz=