Define the function `f(x,y,z)` by `f(x,y,z)=x^(y^(z))-x^(z^(y))+y^(z^(x))-y^(x^(z))+z^(x^(y))`. Evaluate `f(1,2,3)+f(1,3,2)+f(2,1,3)+f(2,3,1)`

Prove that : |{:(1,1,1),(x,y,z),(x^(3),y^(3),z^(3)):}|=(x-y)(y-z)(x+y+z)

Find ((x^(2)-y^(2))^(3) + (y^(2) -z^(2))^(3)+ (z^(2) -x^(2))^(3))/((x-y)^(3) + (y-z)^(3) + (z-x)^(3))

Prove the following : |{:(x,y,z),(x^(2),y^(2),z^(2)),(x^(3),y^(3),z^(3)):}|=|{:(x,x^(2),x^(3)),(y,y^(2),y^(3)),(z,z^(2),z^(3)):}|=xyz(x-y)(y-z)(z-x)

Given x,y>0,z in R. Define f(x,y,z)=((x+y)^(2))/((x sin^(2)z+y cos^(2)z)(x cos^(2)z+y sin^(2)z))+(x)/(y)+(y)/(x) Find minimum value of f(x,y,z)

verify that: x^(3)+y^(3)+z^(3)-3xyz=((1)/(2))(x+y+z)((x-y)^(2)+(y-z)^(2)+(z-x)^(2))

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

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

The equations x+2y+3z=1 2x+y+3z=1 5x+5y+9z=4

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

(x + y) ^ (3) + (y + z) ^ (3) + (z + x) ^ (3) -3 (x + y) (y + z) (z + x)