Prove that `(a^(1/(x-y)))^(1/(x-z))times(a^(1/(y-z)))^(1/(y-x))times(a^(1/(z-x)))^(1/(z-y))=1`

If x,y,z are all different real numbers,then prove that (1)/((x-y)^(2))+(1)/((y-z)^(2))+(1)/((z-x)^(2))=((1)/(x-z)+(1)/(y-z)+(1)/(z-x))^(2)

(1)/((x-y)^(2))+(1)/((y-z)^(2))+(1)/((z-x)^(2))=((1)/(x-y)+(1)/(y-z)+(1)/(z-x))^(2)

Show that: ((a^(x+1))/(a^(y+1)))^(x+y)((a^(y+2))/(a^(z+2)))^(y+z)((a^(z+3))/(a^(x+3)))^(z+x)=1

Prove that sqrt(x^(-1)y)xxsqrt(y^(-1)z)xxsqrt(z^(-1)x)=1

((a^(x^(2)))/a^(y^(2)))^((1)/(x+y))xx((a^(y^(2)))/a^(z^(2)))^((1)/(y+z))xx((a^(Z^(2)))/a^(x^(2)))^((1)/(z+x))= _______

(x^(-1)-y^(-1))/(z)+(y^(-1)-z^(-1))/(x)+(z^(-1)-x^(-1))/(y)= _______

If x + y + z = xyz , prove that x(1 -y^(2)) (1- z^(2))+ y(1- z^(2))(1- x^(2)) +z(1-x^(2)) (1- y^(2)) = 4xyz .