Simplify a 2 - b 2 3 + b 2 - c 2 3 + c 2 - a 2 3 ( a - b ) 3 + ( b - c ) 3 + ( c - a ) 3

Given: a 2 - b 2 3 + b 2 - c 2 3 + c 2 - a 2 3 ( a - b ) 3 + ( b - c ) 3 + ( c - a ) 3 Take the numerator a 2 - b 2 3 + b 2 - c 2 3 + c 2 - a 2 3 . We know that, If x + y + z = 0 , then x 3 + y 3 + z 3 = 3 x y z Here, a 2 - b 2 + b 2 - c 2 + c 2 - a 2 = 0 ⇒ a 2 - b 2 3 + b 2 - c 2 3 + c 2 - a 2 3 = 3 a 2 - b 2 b 2 - c 2 c 2 - a 2 Similarly, take the denominator ( a - b ) 3 + ( b - c ) 3 + ( c - a ) 3 , Here, a - b + b - c + c - a = 0 ⇒ ( a - b ) 3 + ( b - c ) 3 + ( c - a ) 3 = 3 a - b b - c c - a So, a 2 - b 2 3 + b 2 - c 2 3 + c 2 - a 2 3 ( a - b ) 3 + ( b - c ) 3 + ( c - a ) 3 = 3 a 2 - b 2 b 2 - c 2 c 2 - a 2 3 ( a - b ) ( b - c ) ( c - a ) ⇒ a 2 - b 2 3 + b 2 - c 2 3 + c 2 - a 2 3 ( a - b ) 3 + ( b - c ) 3 + ( c - a ) 3 = 3 a + b a - b b + c b - c c + a c - a 3 ( a - b ) ( b - c ) ( c - a ) ⇒ a 2 - b 2 3 + b 2 - c 2 3 + c 2 - a 2 3 ( a - b ) 3 + ( b - c ) 3 + ( c - a ) 3 = a + b b + c c + a Hence, a 2 - b 2 3 + b 2 - c 2 3 + c 2 - a 2 3 ( a - b ) 3 + ( b - c ) 3 + ( c - a ) 3 = a + b b + c c + a