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

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

if omega!=1 is cube root of unity and x+y+z!=0(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

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

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 1, omega, omega^(2) are cube roots of unity, prove that (x + y)^(2) + (x omega + y omega^(2))^(2) + (x omega^(2) + y omega)^(2)= 6xy

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

Prove that (x+y+z)(x+omega y+omega^(2)z)(x+omega^(2)y+omega z)=x^(3)+y^(3)+z^(3)-3xyz