imc

Question

Let denote the "averaged with" operation: a @ b = a+b2. Which of the following distributive laws hold for all numbers x, y, and z? I. x @ (y + z) = (x @ y) + (x @ z) II. x + (y @ z) = (x + y) @ (x + z) III. x @ (y @ z) = (x @ y) @ (x @ z)

Accepted Answer

Simplify each operation and see which ones hold true. align* I. x @ (y + z) &= (x @ y) + (x @ z) x+y+z2 &= x+y2 + x+z2 x+y+z2 &= 2x+y+z2 align* align* II. x + (y @ z) &= (x + y) @ (x + z) x+ y+z2 &= 2x+y+z2 2x+y+z2 &= 2x+y+z2 align* align* III. x @ (y @ z) &= (x @ y) @ (x @ z) x @ y+z2 &= x+y2 @ x+z2 2x+y+z4 &= 2x+y+z4 align* II and III only