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:
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:
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.
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:
We can change the number to the expression
and rewrite the exercise:
Then continue with the distributive property:
You can read about the distributive property of multiplication at this link.
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:
We will round up the number to . To preserve the value of we will write
We will get:
We will divide by and subtract the quotient of divided by
We will get:
Click here to see a more detailed explanation of the commutative property of division.
\( 12:(2\times2)= \)
\( 5\cdot5\cdot5\cdot2\cdot2\cdot2=? \)
\( 7-(4+2)= \)
\( 8-(2+1)= \)
\( 13-(7+4)= \)
According to the order of operations, we first solve the exercise within parentheses:
Now we divide:
We use the substitution property and organize the exercise in the following order:
We place parentheses in the exercise:
We solve from left to right:
1000
According to the order of operations, we first solve the exercise within parentheses:
Now we solve the rest of the exercise:
According to the order of operations, we first solve the exercise within parentheses:
Now we solve the rest of the exercise:
According to the order of operations, we first solve the exercise within parentheses:
Now we subtract:
\( 38-(18+20)= \)
\( 28-(4+9)= \)
\( 55-(8+21)= \)
\( 37-(4-7)= \)
\( 80-(4-12)= \)
According to the order of operations, first we solve the exercise within parentheses:
Now, the exercise obtained is:
According to the order of operations, we first solve the exercise within parentheses:
Now we obtain the exercise:
According to the order of operations, we first solve the exercise within parentheses:
Now we obtain the exercise:
According to the order of operations, we first solve the exercise within parentheses:
Now we obtain:
Remember that the product of a negative and a negative results in a positive, therefore:
Now we obtain:
According to the order of operations, we first solve the exercise within parentheses:
Now we obtain the exercise:
Remember that the product of plus and plus gives us a positive:
Now we obtain:
\( 100-(30-21)= \)
\( 66-(15-10)= \)
\( 22-(28-3)= \)
\( 60:(5\times3)= \)
\( 2+4-3= \)
According to the order of operations, we first solve the exercise within parentheses:
Now we obtain:
According to the order of operations rules, we first solve the expression inside of the parentheses:
We obtain the following expression:
According to the order of operations, we first solve the exercise within parentheses:
Now we obtain the exercise:
We write the exercise in fraction form:
We break down 60 into a multiplication exercise:
We simplify the 3s and obtain:
We break down the 5 into a multiplication exercise:
We simplify the 5 and obtain:
We solve the exercise from left to right, we place the addition exercise in parentheses and then subtract:
3