Question

`(x/2 + y + z/3)^3 + (x/2 + (2y)/3 + z)^3 + (-(5x)/6 - y/3 - (4z)/3)^3`

Answer 1

Let `(x/2 + y + z/3) = a, (x/3 - (2y)/3 + z) = b,  (-(5x)/6 - y/3 - (4z)/3) = c`

`a + b + c = x/2 + y + z/3 + x/3 - (2y)/3 + z - (5x)/6 - y/3 - (4z)/3`

`a + b + c = (x/2 + x/3 - (5x)/6) + (y - (2y)/3 - y/3) + (z/3 + z - (4z)/3)`

`a + b + c = (3x)/6 + (2x)/6 - (5x)/6 + (3y)/3 - (2y)/3 - y/3 + z/3 + (3z)/3 - (4z)/3`

`a + b + c = (5x - 5x)/6 + (3y - 3y)/3 + (4z - 4z)/3`

a + b + c = 0

∵ a + b + c = 0     `∴ a^3 + b^3 + c^3 = 3abc`

`∴ [x/2 + y + z/3]^3 + [x/3 - (2y)/3 + z]^3 + [-(5x)/6 - y/3 - (4z)/3]^3 = 3(x/2 + y + z/3)(x/3 - (2y)/3 + z)(-(5x)/6 - y/3 - (4z)/3)`