The Commutative, Associative and Distributive Laws (or Properties)

In mathematics, there are three main properties that dictate how numbers behave in certain situations: the commutative law or property, the associative law or property, and the distributive law or property. Understanding these properties makes certain mathematical operations much simpler. In this article, we'll focus on the commutative property and demonstrate how it can simplify calculations involving addition and multiplication.

The commutative law or property

The commutative property states that the order in which you add or multiply two real numbers does not affect the result. This property does not apply to subtraction and division.

For addition and multiplication, the commutative property can be expressed as follows:

• Commutative law of addition: $a+b=b+a$
• Commutative law of multiplication: $a\times b=b\times a$

Examples:

Addition

$35+7=42$

$7+35=42$

Addition with negative numbers

$24+(-9)=15$

$(-9)+24=15$

However, the commutative property does not hold for subtraction:

$35-7=28$

$7-35=-28$

Multiplication

$4\times 5=20$

$5\times 4=20$

Multiplication with negative numbers

$13\times \left(-2\right)=-26$

$\left(-2\right)\times 13=-26$

The commutative property does not apply to division:

$26÷2=13$

$2÷26\approx 0.077$

The associative law or property

The associative property states that when you add or multiply any three real numbers, the grouping (or association) of the numbers does not affect the result. The associative property does not apply to subtraction or division of real numbers.

• Associative law of addition: $a+\left(b+c\right)=\left(a+b\right)+c$
• Associative law of multiplication: $a\times \left(b\times c\right)=\left(a\times b\right)\times c$

Examples:

Addition

$3+\left(7+10\right)=3+17=20$

$\left(3+7\right)+10=10+10=20$

Addition with negative numbers

$-5+\left(8+12\right)=-5+20=15$

$\left(-5+8\right)+12=3+12=15$

However, the associative property does not hold for subtraction:

$-5-\left(8-12\right)=-5-\left(-4\right)=-1$

$\left(-5-8\right)-12=-13-12=-25$

Multiplication

$5\times \left(3\times 4\right)=5\times 12=60$

$\left(5\times 3\right)\times 4=15\times 4=60$

Multiplication with negative numbers

$-2\times \left(4\times 7\right)=-2\times 28=-56$

$\left(-2\times 4\right)\times 7=-8\times 7=-56$

The associative property does not apply to division:

$8÷\left(4÷2\right)=8÷2=4$

$\left(8÷4\right)÷2=2÷2=1$

The distributive law or property

The distributive property is an algebraic property used to multiply a single value with two or more values within a set of parentheses. It states that when a factor is multiplied by the sum of two terms, it is possible to multiply each of the two numbers by the factor and then perform the addition operation.

• Distributive property: $a\times \left(b+c\right)=a\times b+a\times c$

Examples:

Using the distributive property: $3\times \left(5+2\right)$

$3\times 5+3\times 2=15+6=21$

$3\times \left(7\right)=21$

Using the distributive property: $5\times \left(-6+8\right)$

$5\times \left(-6\right)+5\times \left(8\right)=-30+40=10$

$5\times \left(2\right)=10$

Topics related to the Commutative, Associative and Distributive Laws (or Properties)

Commutative Property

Associative Property

Basic Operations

Flashcards covering the Commutative, Associative and Distributive Laws (or Properties)

5th Grade Math Flashcards

Common Core: 5th Grade Math Flashcards

Practice tests covering the Commutative, Associative and Distributive Laws (or Properties)

MAP 5th Grade Math Practice Tests

Common Core: 5th Grade Math Diagnostic Tests

Get help learning about the commutative, associative, and distributive properties

Tutoring is an excellent way to help your student become more familiar with the commutative, associative, and distributive properties of addition and multiplication. It can take a while for your student to memorize each of these properties and how to apply them to certain math problems, and some extra 1-on-1 assistance can be a great benefit to your student. With the undivided attention of an expert and a customized learning plan, your student has the best chance of becoming completely familiar with the commutative, associative, and distributive properties and how to use them. We will be happy to set your student up with a tutor who matches their individual needs. Contact Varsity Tutors today to speak with one of our helpful Educational Directors to get started.