﻿ Properties of Multiplication

Properties for Addition of Whole Numbers

Commutative Property of Multiplication:  The Commutative Property of Multiplication of Whole Numbers says that the order of the factors does not change the product.

General Property:  ab = ba

Numeric Example:  3 × 5 = 15 = 5 × 3

Algebraic Example:  (3x)(4x) = (4x)(3x)

Associative Property of Multiplication:  The Associative Property of Multiplication of Whole Numbers says that how the factors are grouped does not change the product.

General Property:  (ab)c = a(bc)

Numeric Example:  (2 × 6) × 8   = 12 × 8

= 96
= 2
× 48
= 2
× (6 ×
8)

Algebraic Example:  2 (3x)  = (2 3)x

= 6x

Notice that in this case, regrouping allows us to simplify the expression.

Example:  We show how the associative and commutative properties for multiplication of whole numbers are used to simplify an algebraic expression.

(3x)(4x) = 3(x 4)x              Associative Property of Multiplication

= 3(4 x) x             Commutative Property of Multiplication

= (3 4)(x x)        Associative Property of Multiplication

= 12x2

Identity Property for Multiplication:  The Identity Property for Multiplication of Whole Numbers says that when a value is multiplied by one the product is that value; i.e., multiplication by one does not change the value of a number. One is called the multiplicative identity.

General Property:  1   a  = a 1 = a

Numeric Example:  1 5  = 5 1 = 5

Algebraic Example:  1(4x) = (4x) 1 = 4x

Distributive Properties of Multiplication:  The Distributive Property of Multiplication over Addition of Whole Numbers (the Distributive Property of Multiplication over Subtraction of Whole Numbers) shows us how multiplying a value times a sum (difference) may be broken into the sum (difference) of separate products.

General Property:   a(b + c) = ab + ac   or  a(b c) = ab ac

Numeric Example:  4(145) = 4(100 + 40 + 5)

= 4(100) + 4(40) + 4(5)

=   400   +   160  +  20

=   580

Algebraic Example:  5(3x + 9)  =  5(3x) + 5(9)

= (5 3)x + 5(9)

=  15x + 45

Note that the Associative Property of Multiplication is used in the second step.

Example:  The distributive property allows us to more easily perform computations mentally.

7(29)  = 7(30 – 1)

= 7(30) – 7(1)

= 210 – 7 = 203 