If a² + b² + c² + 27 = 6(a + b + c), then what is the value of $$\root 3 \of {{a^3} + {b^3} - {c^3}} ?$$

Solution (By Examveda Team)

$$\eqalign{ & {a^2} + {b^2} + {c^2} + 27 = 6\left( {a + b + c} \right) \cr & {\left( {a - 3} \right)^2} + {\left( {b - 3} \right)^2} + {\left( {c - 3} \right)^2} = 0 \cr & a = 3,\,b = 3,\,c = 3 \cr & \root 3 \of {{a^3} + {b^3} - {c^3}} \cr & = \root 3 \of {27 + 27 - 27} \cr & = \root 3 \of {27} \cr & = 3 \cr} $$