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.

Practice Commutative, distributive and associative properties

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

$-3$

Exercise #1

Solve the exercise:

84:4=

Step-by-Step Solution

There are several ways to solve the exercise,

We will present two of them.

In both ways, in the first step we divide the number 84 into 80 and 4.

$\frac{4}{4}=1$

And thus we are left with only the 80.

From the first method, we will decompose 80 into$10\times8$

We know that:$\frac{8}{4}=2$

And therefore, we reduce the exercise $\frac{10}{4}\times8$

In fact, we will be left with$2\times10$

which is equal to 20

In the second method, we decompose 80 into$40+40$

We know that: $\frac{40}{4}=10$

And therefore: $\frac{40+40}{4}=\frac{80}{4}=20=10+10$

which is also equal to 20

Now, let's remember the 1 from the first step and add them:

$20+1=21$

And thus we manage to decompose that:$\frac{84}{4}=21$

21

Exercise #2

$7+8+12=$

Step-by-Step Solution

According to the rules of the order of operations, you can use the substitution property and start the exercise from right to left to calculate comfortably:

$8+12=20$

Now we obtain the exercise:

$7+20=27$

27

Exercise #3

$94+12+6=$

Step-by-Step Solution

According to the rules of the order of operations, you can use the substitution property and organize the exercise in a more convenient way for calculation:

$94+6+12=$

Now, we solve the exercise from left to right:

$94+6=100$

$100+12=112$

112

Exercise #4

$7\times5\times2=$

Step-by-Step Solution

According to the rules of the order of operations, you can use the substitution property and start the exercise from right to left to comfortably calculate:

$5\times2=10$

$7\times10=70$

70

Exercise #5

$3\times5\times4=$

Step-by-Step Solution

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

But, this can leave us with awkward or complicated numbers to calculate.

Since the entire exercise is a multiplication, you can use the associative property to reorganize the exercise:

3*5*4=

We will start by calculating the second exercise, so we will mark it with parentheses:

3*(5*4)=

3*(20)=

Now, we can easily solve the rest of the exercise:

3*20=60

60

Exercise #1

$12\times5\times6=$

Step-by-Step Solution

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

$12\times5=60$

$60\times6=360$

360

Exercise #2

$24:8:3=$

Step-by-Step Solution

According to the order of operations, we solve the exercise from left to right since the only operation in the exercise is division:

$24:8=3$

$3:3=1$

$1$

Exercise #3

$7+4+3+6=\text{?}$

Step-by-Step Solution

To make solving the exercise easier, we try to add numbers that give us a result of 10.

Let's keep in mind that:

$7+3=10$

$6+4=10$

Now, we obtain a more convenient exercise to solve:

$10+10=20$

20

Exercise #4

$19+34+21+10+6=\text{?}$

Step-by-Step Solution

To make solving easier, we try to add numbers that give us a round result.

Keep in mind that:

$19+21=40$

$34+6=40$

Now, we get a more convenient exercise to solve:

$40+40+10=80+10=90$

90

Exercise #5

$74+32+6+4+4=\text{?}$

Step-by-Step Solution

To make solving easier, we try to add numbers that give us a round result.

Keep in mind that:

$4+4=8$

Now we get the exercise:

$74+36+6+8=$

Keep in mind that:

$74+6=80$

$32+8=40$

Now, we get a more comfortable exercise to solve:

$80+40=120$