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

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?

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

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

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

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.

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.

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

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.

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: $508 \times 4=$
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:
$500 \times 4+8 \times 4=$
$2000+32=2032$

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

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

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.

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.