If x = 255, y = 256, z = 257, then find the value of x³ + y³ + z³ - 3xyz.

Solution (By Examveda Team)

$$\eqalign{ & {x^3} + {y^3} + {z^3} - 3xyz \cr & = \frac{{\left( {x + y + z} \right)}}{2}\left[ {{{\left( {x - y} \right)}^2} + {{\left( {y - z} \right)}^2} + {{\left( {z - x} \right)}^2}} \right] \cr & = \frac{{255 + 256 + 257}}{2}\left[ {{1^2} + {1^2} + {2^2}} \right] \cr & = \frac{{768 \times 6}}{2} \cr & = \frac{{4608}}{2} \cr & = 2304 \cr} $$