`(x omega^2+y omega+z)/(x omega+y+omega^2)=((xomega+y+zomega^2)/(xomega^2+yomega+z))^2`

If omega is an imaginary cube root of unity then prove that (xomega^2+yomega+z)/(xomega+y+zomega^2)=((xomega+y+zomega^2)/(xomega^2+yomega+z))^2

If omega be imaginary cube root of 1 , then prove that (xomega^(2)+yomega+z)/(xomega+y+zomega^(2))=((xomega+y+zomega^(2))/(xomega^(2)+yomega+z))^(2)

If 1,omega,omega^2 are the cube roots of unity then prove the following i) (1-omega+omega^2) + (1+omega-omega^2)^5=32 ii) (x+y+z)(x+yomega+zomega^2)(x+yomega^2+zomega)=x^3+y^3+z^3-3xyz

If omega be an imaginary cube root or unity, prove that (x+y omega+ z omega^(2))^(4)+ (x omega+ y omega^(2)+z)^(4)+(x omega^(2)+y+ z omega)^(4)=0

If omega be an imaginary cube root of unity, show that (x+yomega+zomega^(2))^(2)+(xomega+yomega^(2)+z)^(2)+(xomega^(2)+y+zomega)^(2)=0

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

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

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)/(omega^2+1),x/(1+omega),y/(omega+omega^2)]|=0 if x=y=z

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)/(omega^2+1),x/(1+omega),y/(omega+omega^2)]|=0 if x=y=z

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