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Commutative, Associative, and Distributive Properties

As you may have already realized through the years of math classes and homework, math is sequential in nature, meaning that each concept is built upon prior work. Arithmetic skills are necessary to conquer algebraic concepts, which are then developed further to be used in calculus, and so on. As you’re building these concepts over time, the math process may become automatic, but the reason, or justification for the work, may be long forgotten.

In this video, we will go back to the basics to review the commutative, associative, and distributive properties of real numbers, which allow for the math mechanics of algebra and beyond.

Commutative Property

The names of the properties that we’re going to be looking at are helpful in decoding their meanings. Consider the word commutative. What do you think of when you see this word? When I look at this word, I see the word commute. That word reminds me of “move,” which is pretty much what the commutative property allows you to do when adding or multiplying algebraic terms. The commutative property looks like this, mathematically:

Let’s take a minute to remember the definition of an algebraic term: it is the number, variable, or product of coefficients and variables. Examples of algebraic terms are

and so on. To prove that moving, or rearranging, terms is acceptable, let’s look at a few examples of the commutative property being used in addition problems.

Example 1

Example 2

Let’s alter one of our terms a bit for this next example.

Note that there is a very important distinction between the addition of a negative integer and the operation of subtraction. It is important to note this distinction because the commutative property does not apply to the operation of subtraction. For instance,

This property also does not apply to division.

Example 3

The commutative property does, however, apply to multiplication. For example,

Let’s do the math just to make sure.

And

Even though we switched around the terms, we got the same result. Our final example involves the use of variables. Simply substitute values for the variables to show that rearranging terms is acceptable when adding and multiplying.

and z=3

You could substitute with any values, but we’ll use these three for now. So this gives us:

Once we add each side, we are left with 16 on both sides, which is true. 16=16

The next property that we will look at is the associative property.

Associative Property

Again, the name provides a helpful hint to its meaning. What comes to mind when you hear the word associative? For me, the word associate stands out, which could also bring to mind the word group. Accordingly, the associative property allows us to group terms that are joined by addition or multiplication in various ways. Parentheses are used to group the terms, and they establish the order of operations. Work inside the parentheses is always done first. Mathematically, the property looks like this:

Let’s look at an example of this property used in an addition problem.

Example 1

This example will show that adding the last two terms first or adding the first two terms first simply does not matter. Let’s take a look at

So we do what’s in the parentheses first.

So 12=12 because it’s on both sides of the equation. Likewise, the order that we perform multiplication does not matter either.

Example 2

Let’s say we have (3⋅4)⋅5=3⋅(4⋅5).

12⋅5=3⋅20

60=60

The commutative property of multiplication shows that it is acceptable to rearrange terms when multiplying. In contrast, the associative property of multiplication moves parentheses to order the multiplication.

Associative Property of Multiplication

The associative property of multiplication states that the product of three or more numbers remains the same irrespective of the way the numbers are grouped. The associative property of multiplication can be expressed with the help of the following formula:

Associative Law of Multiplication Formula

(A × B) × C = A × (B × C)

Let us understand this with the following example.

Example: (1 × 7) × 3 = 1 × (7 × 3) = 21. When we solve the left-hand side, we get 7 × 3 = 21. Now, when we solve the right-hand side, we get 1 × 21 = 21. Therefore, it can be seen that the product of the numbers remains the same irrespective of the different grouping of numbers.

Associative Property of Subtraction

The associative property does not work with subtraction. This means if we try to apply the associative law to subtraction, it will not work. For example, (7 – 1) – 3 is not equal to 7 – (1 – 3). If we solve the left-hand side, we get, 6 – 3 = 3. Now, if we solve the right-hand side, we get, 7 – (-2) = 9. Hence, we can see there is no associative property of subtraction.

Verification of Associative Law

Let us try to justify how and why the associative property is only valid for addition and multiplication operations. We will apply the associative law individually on the four basic operations.

For Addition: The associative law in Maths for addition is expressed as (A + B) + C = A + (B + C). So, let us substitute this formula with numbers to verify it. For example, (1 + 4) + 2 = 1 + (4 + 2) = 7. Therefore, the associative property is applicable to addition.
For Subtraction: Let us try the associative property formula in subtraction. This can be expressed as (A – B) – C ≠ A – (B – C). Now, let us verify this formula by substituting numbers in this. For example, (1 – 4) – 2 ≠ 1 – (4 – 2) i.e., -5 ≠ -1. Therefore, we say that the associative property is not applicable to subtraction.
For Multiplication: The associative law for multiplication is given as (A × B) × C = A × (B × C). For example, (1 × 4) × 2 = 1 × (4 × 2) = 8. Therefore, we can say that the associative property is applicable to multiplication.
For Division: Now, let us try the associative property formula for division. This can be expressed as (A ÷ B) ÷ C ≠ A ÷ (B ÷ C). For example, (9 ÷ 3) ÷ 2 ≠ 9 ÷ (3 ÷ 2) = 3/2 ≠ 6. Therefore, we can see that the associative property is not applicable to division.

Distributive Property

Finally, the last property we’ll be looking at is the distributive property, which looks like this:

a⋅(b+c)=a⋅b+a⋅c

The notation, once again, dictates that this property applies only to the operations of multiplication and addition. Specifically, if a term is being multiplied by an expression in parentheses, then the multiplication is performed on each of the terms. Here is an example to prove that this algebraic move is justified.

2(3+7)=2⋅3+2⋅7

The parentheses on the left tell us to first add 3+7.

2(10)=6+14
20=20

The sum of the products on the right side of the equation gives the same result as multiplying on the left.

Associative Property of Addition and Multiplication

The associative property is applicable to addition and multiplication, but it does not exist in subtraction and division. We know that the associative property of addition says that the grouping of numbers does not change the sum of a given set of numbers. This means, (7 + 4) + 2 = 7 + (4 + 2) = 13. Similarly, the associative property of multiplication says that the grouping of numbers does not change the product of the given set of numbers. This formula is expressed as (a × b) × c = a × (b × c). For example, (2 × 3) × 4 = 2 × (3 × 4) = 24.

Important Notes:

The associative property is applicable only to addition and multiplication.
Associative properties are in line with the ability to associate or group numbers, which is not possible in the case of subtraction and division.
The associative property is among the list of properties in mathematics that are helpful in the manipulation of mathematical equations and their solutions.

Associative Property Definition

The associative law which applies only to addition and multiplication states that the sum or the product of any 3 or more numbers is not affected by the way in which the numbers are grouped by parentheses. In other words, if the same numbers are grouped in a different way for addition and multiplication, their result remains the same.

The formula for the associative property of addition and multiplication is expressed as:

Let us discuss in detail the associative property of addition and multiplication with examples.

Associative Property of Addition

According to the associative property of addition, the sum of three or more numbers remains the same irrespective of the way the numbers are grouped. Suppose we have three numbers: a, b, and c. For these, the associative property of addition will be expressed with the following formula:

Associative Property of Addition Formula:

The formula for the associative property of addition shows that grouping of numbers in a different way does not affect the sum. The brackets that group the numbers help to make the process of addition simpler. Observe the following formula for the associative property of addition.

(A + B) + C = A + (B + C)

Let us take an example to understand and prove the formula. Let us group 13 + 7 + 3 in three ways.

Step 1: We can group the set of numbers as (13 + 7) + 3, 13 + (7 + 3), and (13 + 3) + 7.
Step 2: Add the first set of numbers, that is, (13 + 7) + 3. This can be further solved as 20 + 3 = 23.
Step 3: Add the second set, i.e., 13 + (7 + 3) = 13 + 10 = 23.
Step 4: Now, solve the third set, i.e., (13 + 3) + 7 = 16 + 7 = 23.
Step 5: The sum of all three expressions is 23. This shows that no matter how we group the numbers with the help of brackets, the sum remains the same.

Example: (1 + 7) + 3 = 1 + (7 + 3) = 11. If we solve the left-hand side, we get, 8 + 3 = 11. Now, if we solve the right-hand side, we get, 1 + 10 = 11. Hence, we can see that the sum remains the same even when the numbers are grouped in a different way.

Review

Ok, now that we’ve gone over the three properties, let’s test your memory. For each problem, state the property, commutative, associative, or distributive, that justifies the statement. Go ahead and pause the video if you need more time.

Problem 1:

5⋅(2⋅x)=(5⋅2)⋅x

Problem 2:

24−3x=3(8−x)

Problem 3:

5×2−2x+3=−2x+3+5×2

Think you got it? Let’s see! The answer for number 1 is the associative property, because the parentheses are moved to order the multiplication. The answer for number two is the distributive property, because 3 is multiplied by both terms in the parentheses. That leaves us with the answer to number three being the commutative property, because we’ve simply rearranged the terms.

Associative Property Examples

Example 1: If 3 × (6 × 4) = 72, then find the product of (3 × 6) × 4 using the associative property.

Solution:Since multiplication satisfies the associative property formula, (3 × 6) × 4 = 3 × (6 × 4) = 72

Example 2: Solve for x using the associative property formula: 2 + (x + 9) = (2 + 5) + 9

Solution:Since addition satisfies the associative property, (2 + 5) + 9 = 2 + (x + 9) = (2 + x) + 9. So, the value of x is 5.

Example 3: If 2 × (3 × 5) = 30, find the product of (2 × 3) × 5 using the associative property.

Solution:The associative property formula is expressed as (A × B) × C = A × (B × C)Given = 2 × (3 × 5) = 30Using the associative property formula, we can evaluate (2 × 3) × 5.To verify: (2 × 3) × 5 = 30 or not, first, let us solve the terms inside parentheses and then multiply it with the number given outside.= 6 × 5= 30Hence, 2 × (3 × 5) = (2 × 3) × 5 = 30.

Q.1. Solve 12 + (15 + 7) using Associative Property of Addition.

Solution:

12 + (15 + 7)
= (12 + 15) + 7
= 27 + 7
= 34

Q.2. Solve 3 × (2 × 6) using Associative Property of Multiplication.

Solution:
3 × (2 × 6)
= (3 × 2) 6
= 6 × 6
= 36

Associative Property of Addition Examples

Example 1: Which equation shows an example of the associative property of addition?a.) (25 + 2) + 8 = 25 + (2 + 8)b.) 7 × (20 – 3) = (7 × 20) – (7 × 3)

Solution:a.) Let us take this equation, (25 + 2) + 8 = 25 + (2 + 8)This equation shows an example of the associative property of addition.b.) Let us take this equation, 7 × (20 – 3) = (7 × 20) – (7 × 3)We can see that this equation shows the distributive property of addition.

Example 2: Fill in the missing number and then write the sum:7 + (10 + 6) = (7 + 10) + ___ = ___

Solution: According to the associative property of addition formula, a + (b + c) = (a + b) + c. If we substitute the values in this formula we get 6 as the missing number, that is, 7 + (10 + 6) = (7 + 10) + 6, and the sum is 23.

Example 3: Choose the correct option for the missing number.8 + (4 + 2) = (8 + ___) + 2a) 4b) 7c) 6

Solution:According to the associative property of addition: a + (b + c) = (a + b) + c. Substituting the values in the formula: 8 + (4 + 2) = (8 + 4) + 2.Hence, the missing number is 4 because the sum of both the expressions is 14.Therefore, the correct option is (a).

Frequently Asked Questions

What is the commutative property in math?

The commutative property applies to addition and multiplication. The property states that terms can “commute,” or move locations, and the result will not be affected. This is expressed as a+b=b+a for addition, and a×b=b×a for multiplication. The commutative property does not apply to subtraction or division.

What are 2 examples of the commutative property?

The commutative property applies to addition and multiplication. For example, if you have 4 coins in your left pocket and 5 coins in your right pocket, you have 9 coins in all, regardless of which pocket you count first.

a+b=b+a
4+5=5+4

The same concept is true for multiplication. For example, in an ice cube tray with 2 rows of 10 cubes, there will be 20 cubes in all, regardless of how you count the cubes. Counting 2 rows of 10, or counting 10 rows of 2 will both generate the same result.

a×b=b×a
2×10=10×2

How do you verify the commutative property?

The commutative property can be verified using addition or multiplication. This is because the order of terms does not affect the result when adding or multiplying.

For example, when multiplying 5 and 7, the order does not matter. (5)×(7)=35

and (7)×(5)=35.

Multiplying 5 chairs per row by 7 rows will give you 35 chairs total, and multiplying 7 chairs per row by 5 rows will also give you 35 chairs total.

Similarly, the order of terms is irrelevant when adding. For example,

(5)+(7)=12 and (7)+(5)=12.

If I add 7 blue gumballs to 5 red gumballs, I will have 12 gumballs total. And if I add 5 blue gumballs to 7 red gumballs, I will still have 12 gumballs total.

Is division a commutative property?

The commutative property does not apply to division. For example, 500÷2=250, but 2÷500=0.004. When the terms “commute”, or change locations, the answer changes. In division, the order of the terms matters.

What is an associative property example?

The associative property states that when three or more numbers are added or multiplied, and grouping symbols are used, the result will not be affected regardless of where the grouping symbols are located. For example, if you have 5 green marbles, 9 yellow marbles, and 4 blue marbles, you have 18 marbles in all, regardless of which two colors you combine first.

(a+b)+c=a+(b+c)
(5+9)+4=5+(9+4)
(14)+4=5+(13)
18=18

Similarly, the grouping symbols are somewhat arbitrary when multiplying as well. For example, when calculating the volume of a rectangular prism with a length of 5 in, a width of 4 in, and a height of 3 in, the order that you multiply in does not affect the result. Multiplying the length and the width, and then the height, will produce the same result as multiplying the width and the height, and then the length.

(a×b)×c=a×(b×c)
(5×4)×3=5×(4×3)
(20)×3=5×(12)
60=60

What is the associative property formula?

The associative property states that when adding or multiplying, the grouping symbols can be relocated without affecting the result. The formula for addition states (a+b)+c=a+(b+c) and the formula for multiplication states (a×b)×c=a×(b×c).

What is the difference between the associative property and the distributive property?

The associative property states that when adding or multiplying, the grouping symbols can be rearranged and it will not affect the result. This is stated as (a+b)+c=a+(b+c). The distributive property is a multiplication technique that involves multiplying a number by all the separate addends of another number. This is stated as a(b+c)=ab+ac.

What is the distributive property in math?

The distributive property is a method of multiplication where you multiply each addend separately. For example, instead of multiplying 5×46, we can break 46 apart into separate addends (40+6), and multiply 5 by each part separately. 5×46 becomes 5×40 plus 5×6. Essentially the 5 is being “distributed” to each addend.

The distributive property is often used in algebra when simplifying expressions or equations. For example, the distributive property can be used to simplify the expression 4x(2×2+6x−9).

The term 4x will be “distributed” (multiplied) by each term inside the parentheses.

4x×2x²=8x³, 4x×6x=24x², and 4x×−9=−36x. This means that when 4x(2x²+6x−9) is “distributed”, the result is 8x³+24x²−36x.

The distributive property can be defined as a(b+c)=ab+ac.

How do you explain the distributive property?

The distributive property is formally defined as a(b+c)=ab+ac
. The term “distributive property” stems from the term “distribute”. Essentially one number will be “distributed”, or multiplied, by another number that is broken up into separate addends. For example, 6×84
can be solved by “distributing” (multiplying) 6×80
plus 6×4
. This property is widely used in algebra when simplifying expressions or equations. For example, 2x(3x+5)
can be simplified by distributing (multiplying) the term 2x
by each term inside the parentheses 2x×3x=6x²
, and 2x×5=10x
. This means that 2x(3x+5)
simplifies to 6x²+10xz

What is the commutative property formula?

The commutative property formula applies to addition and multiplication. The addition formula states that a+b=b+a, and the multiplication formula states that a×b=b×a. These formulas are used to describe the concept that when adding or multiplying, terms can “commute”, or relocate, and the result will not change.

What is the distributive property in 3rd grade math?

The distributive property is a helpful technique for multiplying multi-digit numbers. For example, 3×4,562
can seem like a daunting task at first glance. However, if you break apart 4,562 into 4,000+500+60+2
, the pieces become much more manageable. We can now multiply 3 by each of these “pieces”. The distributive property often makes multi-digit multiplication much more manageable.

“Distribute” the 3 to all the addends (multiply).

3×4,000=12,000

3×500=1,500

3×60=180

3×2=6
Now add up the pieces. The total is 13,686.

What is the Associative Property in Math?

The associative property or the associative law in math is the property of numbers according to which, the sum or the product of three or more numbers does not change if they are grouped in a different way. In other words, if we add or multiply three or more numbers we will obtain the same answer irrespective of the order of the parentheses. The associative property in math is only applicable to two primary operations, that is, addition and multiplication.

What is the Associative Property of Addition?

The associative property formula of addition says thatthe sum of three or more numbers remains the same irrespective of the way the numbers are grouped. The associative property formula which applies to addition is expressed as (A + B) + C = A + (B + C).

What is the Associative Property of Multiplication?

The associative property formula for multiplication says that the product of three or more numbers remains the same irrespective of the way the numbers are grouped. The associative property formula for multiplication is expressed as (A × B) × C = A × (B × C).

What is the Associative Property Formula for Rational Numbers?

The associative property formula for rational numbers can be expressed as (A + B) + C = A + (B + C) in case of addition, and, (A × B) × C = A × (B × C) in case of multiplication. Here, the values of A, B, and C are in form of p/q, where q ≠ 0. The associative property formula is only valid for addition and multiplication.

Which Two Operations Satisfy the Condition of the Associative Property?

The two operations which satisfy the condition of the associative property are addition and multiplication. This means that the associative property is applicable to addition and multiplication.

Give an Example of the Associative Property of Multiplication.

The associative property of multiplication can be understood with the help of an example. Let us multiply any three numbers (4 × 6) × 10, we get the product as 24 × 10 = 240. Let us group these numbers as 4 × (6 × 10), we still get the product as 4 × 60 = 240. This verifies the associative property of multiplication according to which the product of the numbers remains the same even if they are grouped in a different way.

What is an Example of the Associative Law of Addition?

The associative law of addition can be understood with the help of an example of any three numbers. Let us add (4 + 2) + 10, we get the sum as 6 + 10 = 16. Now, if we group these numbers as 4 + (2 + 10), we still get the sum as 4 + 12 = 16. This proves the associative property of addition which states that the sum of the numbers remains the same even if they are grouped in a different way.

How is the Associative Property Different From the Commutative Property?

The associative property states that the sum or the product of three or more numbers does not change if they are grouped in a different way. This associative property is applicable to addition and multiplication. It is expressed as, (A + B) + C = A + (B + C) and (A × B) × C = A × (B × C). The commutative property states that changing the order of the operands does not change the result of the arithmetic operation. This commutative property is applicable to addition and multiplication. It is expressed as, A × B = B × A and A + B = B + A.

What is the Associative Law and the Distributive Law?

The Associative law states that no matter how we group the numbers in addition and multiplication, the sum or the product remains the same. For example, if we add (5 + 7) + 10, we get 22. Now if we change the grouping of the numbers as 5 + (7 + 10), we still get 22. This is what the Associative law states. According to the Distributive law, an expression that is given in the form of A (B + C) can be solved as A × (B + C) = AB + AC. This distributive law is also applicable to subtraction and is expressed as, A (B – C) = AB – AC. This means operand A is distributed between the other two operands.

How does the Associative Law work?

The Associative law is applicable to addition and multiplication. It says that even if the grouping of numbers is changed, that does not affect the sum or the product. For example, if we multiply 5 × (2 × 3), we get 5 × (6) = 30. Now, if we group the numbers as (5 × 2) × 3, we again get (10) × 3 = 30. Now, let us apply this law to addition. For example, if we add 8 + (3 + 4), we get 15. Now, if we change the grouping of these numbers as (8 + 3) + 4, we still get 15. This is how the Associative law works on addition and multiplication.

Is there any Associative Property of Division?

No, the associative property is not applicable to division and subtraction. Let us try the associative property formula for division. This can be expressed as (A ÷ B) ÷ C ≠ A ÷ (B ÷ C). For example, (12 ÷ 6) ÷ 2 ≠ 12 ÷ (6 ÷ 2). Therefore, we can see that the associative property is not applicable to division. The associative law is only applicable to addition and multiplication.

What is the Benefit of Using the Associative Law of Addition?

The benefit of the associative law of addition is that it helps to form smaller components and this makes the calculation of addition simpler. The grouping of numbers with the help of brackets eases the process of simplifying an expression.

How to Verify the Associative Property of Addition?

The associative law of addition can be easily verified by adding the given set of numbers. For example, let us group 6 + 7 + 8 in two ways.

Step 1: We can group the given set of numbers as (6 + 7) + 8 and 6 + (7 + 8).
Step 2: Now, let us add the first set of numbers, i.e., (6 + 7) + 8. This results in 13 + 8 = 21.
Step 3: Now, let us add the second set, i.e., 6 + (7 + 8) = 6 + 15 = 21.
Step 4: The sum of both the expressions is 21. This proves the associative property of addition which shows that no matter how we group the numbers with the help of brackets, the sum remains the same.

Does the Associative Property of Addition Always Involve 3 or more Numbers?

Yes, the associative property of addition always involves 3 or more numbers because the property rule states that changing the grouping of addends does not change the sum and in the case of only two numbers we cannot make groups.

What is the Formula for the Associative Property of Addition?

The formula for the associative property of addition states that the sum of three or more numbers remains the same no matter how the numbers are grouped. It is expressed as, a + (b + c) = (a + b) + c = (a + c) + b.

What is the Difference Between the Commutative and Associative Property of Addition?

The following points show the difference between the commutative and the associative property of addition:

The commutative property of addition states that changing the order of the addends does not change the sum. For example, 4 + 6 = 6 + 4 = 10. The associative property of addition states that the grouping of numbers does not change the sum. For example, 8 + (2 + 3) = (8 + 2) + 3 = 13.
The commutative property of addition can be applied to two numbers, but the associative property is applicable to 3 or more numbers.
In the commutative property of addition, the order of the addends does not matter, while in the associative property of addition, the grouping of the addends does not matter.

How is the Associative Property of Addition Used in Everyday Life?

There are many places where we can apply the associative property of addition. For example, if we spend $3 on a cupcake, $6 on ice cream, and $2 on candy, we can add up the cost of the items in any order as, 3 + (6 + 2), or, (3 + 6) + 2. Both the expressions result in the same sum, that is, 11. This shows the associative property of addition which says that no matter how we group 3 or more numbers, the sum remains the same.