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

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

If omega!=1 is a cube root of unity and x+y+z!=0, then prove that |x/(1+omega)y/(omega+omega^2)z/(omega^2+1)y/(omega+omega^2)z/(omega^2+1)x/(1+omega)(z z)/(omega^2+1)x/(1+omega)y/(omega+omega^2)|=0 if x=y=z

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

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

(1-omega+omega^(2))(1+omega-omega^(2))=4

If omega is a cube root of unity, prove that (1+omega-omega^2)^3-(1-omega+omega^2)^3=0

If omega is non-real cube roots of unity,then prove that |(x+y omega+z omega^(2))/(xw+z+y omega^(2))|=1

omega is complex root of unity then prove that (2-omega)(2-omega^2)=7

If 1, omega, omega^2 be three roots of 1, show that: (1-omega+omega^2)^2+(1+omega-omega^2)^2=-4

If omega is a complex cube root of unity,show 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)]=[000]