Verify that ` x ^(3) + y ^(3) + z ^(3) - 3xyz =1/2 (x + y + z) [(x-y)^(2) + (y-z) ^(2) + (z-x) ^(2) ]`

If x+y+z=0 show that x^(3)+y^(3)+z^(3)=3xyz .

If x+y+z=0, show that x^3 + y^3+z^3- 3xyz .

If u = log (x^(3) + y^(3) + z^(3) - 3xyz) , then (x + y + z)(u_(x) + u_(y) + u_(z)) is equal to

If 1 , omega , omega^(2) are the cube roots of unity, then prove that ( x + y + z) ( x + y omega + z omega^(2)) ( x + yomega^(2) + z omega) = x^(3) + y^(3) + z^(3) - 3xyz

11x y^2 z^3 // 55xyz = ___

If x+y+z=xyz , " then " sum(3x-x^(3))/(1-3x^(2)) =

If Tan^(-1)x+Tan^(-1)y+Tan^(-1)z=(pi)/2 and (x-y)^(2)+(y-z)^(2)+(z-x)^(2)=0 then x^(2)+y^(2)+z^(2)=

The centre and radius of the sphere 3x^(2) + 3y^(2) + 3z^(2) - 2x - 12y + 6z + 7 = 0 are

If (x_(1), y_(1), z_(1)), (x_(2), y_(2), z_(2)) and (x_(3), y_3), z_(3)) are the vertices of an equilateral triangle such that (x_(1)-2)^(2)+(y_(1)-3)^(2)+(z_(1)-4)^(2)=(x_(2)-2)^(2)+(y_(2)-3)^(2)+(z_(2)-4)^(2)=(x_(3)-2)^(3)+(y_(3)-3)^(2)+(z_(3)-4)^(2) then sum x_(1)+2sumy_(1)+3sum z_(1)=