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

If x=log_(2a) a,y=log_(3a) 2a and z=log_(4a) 3a then prove that xyz+1=2yz

If p=a+bomega+comega^2 , q=b+comega+aomega^2 , and r=c+aomega+bomega^2 , where a ,b ,c!=0 and omega is the complex cube root of unity, then (a) p+q+r=a+b+c (b) p^2+z^2+r^2=a^2+b^2+c^2 (c) p^2+z^2+r^2=-2(p q+q r+r p) (d) none of these

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

Let omega be the imaginary cube root of unity and (a+bomega + comega^2)^(2015) =(a+bomega^2 + c omega) where a,b,c are unequal real numbers . Then the value of a^2+b^2+c^2-ab-bc-ca equals.

omega is an imaginary root of unity. Prove that (i) ( a + bomega + comega^(2))^(3) + (a+bomega^(2) + comega)^(3) = (2a-b-c)(2b -a -c)(2c -a-b) (ii) If a+b+c = 0 then prove that (a + bomega + comega^(2))^(3)+(a+bomega^(2) + comega)^(3) = 27abc .

If |(x, x^2, x^3 +1), (y, y^2, y^3+1), (z, z^2, z^3+1)| = 0 and x ,y and z are not equal to any other, prove that, xyz = -1

If a=z_1+z_2+z_3, b=z_1+omega z_2+omega^2z_3,c=z_1+omega^2z_2+omegaz_3(1,omega, omega^2 are cube roots of unity), then the value of z_2 in terms of a,b, and c is (A) (aomega^2+bomega+c)/3 (B) (aomega^2+bomega^2+c)/3 (C) (a+b+c)/3 (D) (a+bomega^2+comega)/3

If a^x=b^y=c^z\ a n d\ b^2=a c , prove that y=(2x z)/(x+z)