Commutative, Distributive and Associative Properties - Examples, Exercises and Solutions

The commutative properties of addition and multiplication, the distributive property, and many more!

In this article, we will summarize all the basic rules of mathematics that will accompany you in every exercise - the commutative property of addition, the commutative property of multiplication, the distributive property, and all the others!
Shall we begin?

Commutative property

The commutative property can be found in two cases, with addition and with multiplication.
You can read general features of the commutative property at this link.

Commutative property of addition

Thanks to it, we can change the place of the addends without altering the result.
The property is also valid in algebraic expressions.

Rule:
a+b=b+aa+b=b+a

x\cdotnumber~any=number~any\cdot x

Click here to see a more detailed explanation about the commutative property of addition.


Commutative property of multiplication

Thanks to it, we can change the place of the factors without altering the product.
The property is also valid in algebraic expressions.
Rule:
a×b=b×aa \times b=b \times a

x\cdotnumber~any=number~any\cdot x
Click here to see a more detailed explanation about the commutative property of multiplication.

Distributive Property

In the same way, the commutative property can also be found in two cases, with division and with multiplication.
You can read general features of the distributive property at this link.


Distributive property of multiplication

It allows us to distribute - it separates an exercise with several numbers and multiplication operations into another simpler one that has numbers and addition or subtraction operations without changing the result.
The property is also valid in algebraic expressions.

The basic rule:

a(b+c)=ab+aca(b+c)=ab+ac

Multiply the number outside the parentheses by the first number inside the parentheses and, to this product, add or subtract - according to the sign of the exercise - the product of the number outside the parentheses with the second number inside the parentheses.

Additionally
The distributive property allows us to make small changes to the numbers in the exercise to round them as much as possible, making the exercise easier.
For example:
In the exercise: 508×4=508 \times 4=
We can change the number 508508 to the expression (500+8)(500+8)
and rewrite the exercise:
(500+8)×4=(500+8) \times 4=
Then continue with the distributive property:
500×4+8×4=500 \times 4+8 \times 4=
2000+32=20322000+32=2032

You can read about the distributive property of multiplication at this link.


The extended rule

(a+b)(c+d)=ac+ad+bc+bd(a+b)(c+d)=ac+ad+bc+bd

We will choose the expression that is inside the parentheses - we will take one element at a time and multiply it in the given order by each of the elements in the second expression, keeping the subtraction and addition signs.
Then we will do the same with the second element of the chosen expression.

You can read about the extended distributive property right here.


Distributive property of division

Thanks to it, we can round the number we want to divide, always taking into account that the rounded number can actually be divided by the other.
This is done without affecting the original number to preserve its value.

For example:
76:4=76:4=
We will round up the number 7676 to 8080. To preserve the value of 7676 we will write 80480-4
We will get:
(804):4=(80-4):4=
We will divide 8080 by 44 and subtract the quotient of 44 divided by 44
We will get:
80:44:4=80:4-4:4=
201=1920-1=19

Click here to see a more detailed explanation of the commutative property of division.


Practice Commutative, Distributive and Associative Properties

Examples with solutions for Commutative, Distributive and Associative Properties

Exercise #1

15×2×8= 15\times2\times8=

Video Solution

Step-by-Step Solution

Since the exercise involves only multiplication, we will use the commutative property to simplify the calculation:

2×15×8= 2\times15\times8=

Now let's solve the multiplication on the right:

15×8=120 15\times8=120

Now we get the expression:

2×120=240 2\times120=240

Answer

240 240

Exercise #2

5+14+5= 5+14+5=

Video Solution

Step-by-Step Solution

Since the exercise only involves addition, we will use the commutative property to calculate more conveniently:

5+5+14= 5+5+14=

We will solve the exercise from left to right:

5+5=10 5+5=10

10+14=24 10+14=24

Answer

24

Exercise #3

8+9+2= 8+9+2=

Video Solution

Step-by-Step Solution

Since the exercise only involves addition, we will use the commutative property to solve it more conveniently:

8+2+9= 8+2+9=

Let's solve the exercise from left to right:

8+2=10 8+2=10

10+9=19 10+9=19

Answer

19

Exercise #4

13+5+5= 13+5+5=

Video Solution

Step-by-Step Solution

First, we'll solve the right-hand exercise, since adding the numbers will give us a round number:

5+5=10 5+5=10

Now we'll get an easier exercise to solve:

13+10=23 13+10=23

Answer

23

Exercise #5

38+2+8= 38+2+8=

Video Solution

Step-by-Step Solution

First, we'll solve the right-hand exercise, since adding the numbers will give us a round number:

2+8=10 2+8=10

Now we'll get an easier exercise to solve:

38+10=48 38+10=48

Answer

48

Exercise #6

8+2+7= 8+2+7=

Video Solution

Step-by-Step Solution

First, we'll solve the left exercise, since adding the numbers will give us a round number:

8+2=10 8+2=10

Now we'll get an easier exercise to solve:

10+7=17 10+7=17

Answer

17

Exercise #7

13+7+100= 13+7+100=

Video Solution

Step-by-Step Solution

First, we'll solve the left exercise, since adding the numbers will give us a round number:

13+7=20 13+7=20

Now we'll get an easier exercise to solve:

20+100=120 20+100=120

Answer

120

Exercise #8

13+2+8= 13+2+8=

Video Solution

Step-by-Step Solution

We will use the commutative property and first solve the addition exercise on the right:

2+8=10 2+8=10

Now we get:

13+10=23 13+10=23

Answer

23

Exercise #9

4+9+8= 4+9+8=

Video Solution

Step-by-Step Solution

Let's break down 4 into a smaller addition problem:

3+1 3+1

Now we'll get the exercise:

3+1+9+8= 3+1+9+8=

Since the exercise only involves addition, we'll use the commutative property and start with the exercise:

1+9=10 1+9=10

Now we'll get the exercise:

3+10+8= 3+10+8=

Let's solve the exercise from right to left:

10+8=18 10+8=18

18+3=21 18+3=21

Answer

21

Exercise #10

12:(2×2)= 12:(2\times2)=

Video Solution

Step-by-Step Solution

According to the order of operations, we first solve the exercise within parentheses:

2×2=4 2\times2=4

Now we divide:

12:4=3 12:4=3

Answer

3 3

Exercise #11

555222=? 5\cdot5\cdot5\cdot2\cdot2\cdot2=?

Video Solution

Step-by-Step Solution

We use the substitution property and organize the exercise in the following order:

5×2×5×2×5×2= 5\times2\times5\times2\times5\times2=

We place parentheses in the exercise:

(5×2)×(5×2)×(5×2)= (5\times2)\times(5\times2)\times(5\times2)=

We solve from left to right:

10×10×10= 10\times10\times10=

(10×10)×10= (10\times10)\times10=

100×10=1000 100\times10=1000

Answer

1000

Exercise #12

7(4+2)= 7-(4+2)=

Video Solution

Step-by-Step Solution

According to the order of operations, we first solve the exercise within parentheses:

4+2=6 4+2=6

Now we solve the rest of the exercise:

76=1 7-6=1

Answer

1 1

Exercise #13

8(2+1)= 8-(2+1)=

Video Solution

Step-by-Step Solution

According to the order of operations, we first solve the exercise within parentheses:

2+1=3 2+1=3

Now we solve the rest of the exercise:

83=5 8-3=5

Answer

5 5

Exercise #14

13(7+4)= 13-(7+4)=

Video Solution

Step-by-Step Solution

According to the order of operations, we first solve the exercise within parentheses:

7+4=11 7+4=11

Now we subtract:

1311=2 13-11=2

Answer

2 2

Exercise #15

38(18+20)= 38-(18+20)=

Video Solution

Step-by-Step Solution

According to the order of operations, first we solve the exercise within parentheses:

18+20=38 18+20=38

Now, the exercise obtained is:

3838=0 38-38=0

Answer

0 0