Distributive Property (mathematics).

Distributive Property (mathematics). Distributive Property (mathematics), trait of two mathematical operations that allows one operation to be distributed across the other. According to the general distributive law, for example, multiplication may be distributed across addition without changing the value of the overall expression. This law states that x(y + z) = xy + xz for any numbers x, y, and z. For instance, to find the value of the expression 5(3 + 4), add 3 plus 4 to get 7, and then multiply 5 times 7 to get 35. The distributive property allows the expression to be rewritten, with the multiplication by 5 distributed to each of the terms that were added together: 5(3 + 4) = (5 × 3) + (5 × 4). Multiplying 5 times 3 gives 15, and 5 times 4 gives 20. Adding 15 and 20 gives 35, the same value as that of the first calculation. The distributive property works no matter how many instances of the second operation are involved: x(w + y + z) = xw + xy + xz, x(v + w + y + z) = xv + xw + xy + xz, and so on. The distributive law is useful for simplifying algebraic expressions such as: Applying the distributive property to the top part of the expression yields: The y + 3 terms cancel each other because (y + 3)/(y + 3) = 1, leaving x = 7. Multiplication also distributes over subtraction: x(y - z) = xy - xz. For example, 6(4 - 2) = (6 × 4) - (6 × 2) = 12. Not all operations, however, are distributive. Addition and subtraction, for example, do not distribute over multiplication: 6 + (4 × 2) = 14, while (6 + 4)(6 + 2) = 80. See also Arithmetic; Associative Property; Commutative Property. Microsoft ® Encarta ® 2009. © 1993-2008 Microsoft Corporation. All rights reserved.