If `x=a+b,y=aomega+bomega^2, z=aomega^2+bomega`, prove that `x^3+y^3+z^3=3(a^3+b^3)`

If =a+b,y=aomega+bomega^2 and z=aomega^2+bomega, prove tht xyz=a^3+b^3

If x=a+b, y=aomega+bomega^2 and z=aomega^2+bomega where omega is an imaginary cube root of unity, prove that x^2+y^2+z^2=6ab .

If x+y=z prove that x^(3)+y^(3)+3xyz=z^(3)

Let omega =e^(ipi/3) and a,b,c,x,y,z be nonzero complex number such that a+b+c=x, a+bomega+comega^2=y, a+bomega^2+comega=z . Then the value of (|x|^2+|y|^2+|z|^2)/(|a|^2+|b|^2+|c|^2) is

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

Prove that a^3 + b^3 + c^3 – 3abc = (a + b + c) (a + bomega + comega^2) (a + bomega^2 + "c"omega) , where omega is an imaginary cube root of unity.

If x=a+b,y=b+c,z=c+a , prove that (x+y+z)xx{x^(2)+y^(2)+z^(2)-xy-yz-zx}=2(a^(3)+b^(3)+c^(3)-3abc)