The commutative properties of addition and multiplication, and the distributive property

πPractice commutative, distributive and associative properties

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.

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

Rule:
$a+b=b+a$

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

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:
$a \times b=b \times a$

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

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:

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

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.

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: $508 \times 4=$
We can change the number $508$ to the expression $(500+8)$
and rewrite the exercise:
$(500+8) \times 4=$
Then continue with the distributive property:
$500 \times 4+8 \times 4=$
$2000+32=2032$

The extended rule

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

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.

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:
$76:4=$
We will round up the number $76$ to $80$. To preserve the value of $76$ we will write $80-4$
We will get:
$(80-4):4=$
We will divide $80$ by $4$ and subtract the quotient of $4$ divided by $4$
We will get:
$80:4-4:4=$
$20-1=19$

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

Test yourself on commutative, distributive and associative properties!

$$3+2-11=$$

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

Other arithmetic rules:

In specific cases, other rules are used in the aforementioned properties that allow us to obtain shorter results.

Subtraction of a sum

It is valid when we need to subtract a sum of elements and not just a single element.
Rule:
$aβ(b+c)=aβbβc$

We can start by calculating the sum inside the parentheses and then subtract it.
However, we can also apply the subtraction to each of the elements inside the parentheses.
For example:
$15-(4+3)=$

Option 1 β First deal with the parentheses:

$15-7=8$

Option 2 β Apply the subtraction operation to each of the elements included in the parentheses.

$15-4-3=8$

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Subtraction of a difference

It is valid when we need to subtract a difference of elements and not just a single element.
Rule:
$aβ(b-c)=a-b+c$

We can start by working within the parentheses - calculate the difference and only then subtract it.
However, we can also apply the subtraction to each of the elements included in the parentheses and remember that minus times minus is plus.

Division by product

It is valid when we need to divide a product (multiplication of elements) and not just a single element.
Rule:
$a:(b\cdot c)=a:b:c$
We can start by dealing with the parentheses - calculate the product and only then divide by it.
However, we can also apply the division to each of the elements included in the parentheses.

Do you know what the answer is?

Division by quotient

It is valid when we need to divide a quotient (division of elements) and not just a single element.
Rule: $a:(b:c)=a:b\cdot c$
We will divide the first element of the parentheses and then place a multiplication sign before the second element of the parentheses.

Likewise, we can reach the quotient within the parentheses and then apply the division.

Examples and exercises with solutions of the commutative properties, multiplication, the distributive property, and more!

Exercise #1

$3+2-11=$

Step-by-Step Solution

According to the order of operations, we solve the exercise from left to right:

$3+2=5$

$5-11=-6$

$-6$

Exercise #2

$4+5+1-3=$

Step-by-Step Solution

According to the order of operations, we solve the exercise from left to right:

$4+5=9$

$9+1=10$

$10-3=7$

7

Exercise #3

Solve:

$2-3+1$

Step-by-Step Solution

We use the substitution property and add parentheses for the addition operation:

$(2+1)-3=$

Now, we solve the exercise according to the order of operations:

$2+1=3$

$3-3=0$

0

Exercise #4

Solve:

$3-4+2+1$

Step-by-Step Solution

We will use the substitution property to arrange the exercise a bit more comfortably, we will add parentheses to the addition operation:
$(3+2+1)-4=$
We first solve the addition, from left to right:
$3+2=5$

$5+1=6$
And finally, we subtract:

$6-4=2$

2

Exercise #5

Solve:

$-5+4+1-3$

Step-by-Step Solution

According to the order of operations, addition and subtraction are on the same level and, therefore, must be resolved from left to right.

However, in the exercise we can use the substitution property to make solving simpler.

-5+4+1-3

4+1-5-3

5-5-3

0-3

-3

$-3$