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=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=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 .

18.Prove that det[[b,a,cb,a,cc,b,a]]=(a+b+c)(a+b omega+c omega^(2))(a+b omega^(2)+c omega): where ,omega is a nonreal cube root of unity.

(a+bomega+comega^(2))/(c+aomega+bomega^(2))

The value of (a+bomega+comega^2)/(b+comega+aomega^2)+(a+bomega+comega^2)/(c+aomega+bomega^2) (where 'omega' is the imaginary cube root of unity), is (a) -omega (b). omega^2 (c). 1 (d). -1

The value of (a+bomega+comega^2)/(b+comega+aomega^2)+(a+bomega+comega^2)/(c+aomega+bomega^2) (where 'omega' is the imaginary cube root of unity), is a.-omega b. omega^2 c. 1 d. -1

If x= a+ b, y = a omega + b omega^(2), z= a omega^(2) + b omega , then the value of x^(3) + y^(3) + z^(3) is equal to (where omega is imaginary cube root of unity)