The Commutative Property of Multiplication

πŸ†Practice commutative property

The commutative property of multiplication tells us that changing the order of factors in an expression doesn't change the answer - even if there are more than two!
Like the commutative property of addition, the commutative property of multiplication helps us simplify basic expressions, algebraic expressions and more.


We can define the commutative property of multiplication as:
aΓ—b=bΓ—a a\times b=b\times a

And in algebraic expressions:
XΓ—numero=numeroΓ—XX\times numero=numero\times X

A - The Commutative Property of Multiplication

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Test yourself on commutative property!

einstein

\( 74+32+6+4+4=\text{?} \)

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Let's see an example:
XΓ—2=2Γ—X X\times2=2\times X

We can replace the X with a number:

X=3 X=3

We get:

2Γ—3=6 2\times3=6

3Γ—2=6 3\times2=6

As you can see, it doesn't matter in what order we multiply the factors, we still get the same correct answer.
Note that the commutative property of multiplication does not work with division.

Starting to get it? Keep practicing to turn this new property into muscle memory.


Exercises with the commutative property of multiplication

Exercise 1

Task:

5.25β‹…23β‹…721=?5.25\cdot\frac{2}{3}\cdot\frac{7}{21}=\text{?}

Solution:

First, we convert the decimal number into a mixed fraction.

514β‹…23β‹…721= 5\frac{1}{4}\cdot\frac{2}{3}\cdot\frac{7}{21}=

Then, we convert the mixed fraction into a simple fraction.

5Γ—4+14β‹…23β‹…721=\frac{5\times4+1}{4}\cdot\frac{2}{3}\cdot\frac{7}{21}=

214β‹…23β‹…721= \frac{21}{4}\cdot\frac{2}{3}\cdot\frac{7}{21}=

Notice that you can reduce to 21 and get a simpler expression.

74β‹…23= \frac{7}{4}\cdot\frac{2}{3}=

We multiply the numerator by the numerator and the denominator by the denominator.

7Γ—24Γ—3= \frac{7\times2}{4\times3}=

1412= \frac{14}{12}=

We continue to reduce as much as possible.

1212= 1\frac{2}{12}=

116 1\frac{1}{6}

Answer:

116 1\frac{1}{6}


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Exercise 2

Task:

414β‹…349β‹…3117=?4\frac{1}{4}\cdot3\frac{4}{9}\cdot3\frac{1}{17}=\text{?}

Solution:

First, we convert all mixed fractions to simple fractions.

4Γ—4+14Γ—3Γ—9+49Γ—3Γ—17+117=\frac{4\times4+1}{4}\times\frac{3\times9+4}{9}\times\frac{3\times17+1}{17}=

16+14Γ—27+49Γ—51+117=\frac{16+1}{4}\times\frac{27+4}{9}\times\frac{51+1}{17}=

174Γ—319Γ—5217= \frac{17}{4}\times\frac{31}{9}\times\frac{52}{17}=

We reduce to 17 17

524Γ—319= \frac{52}{4}\times\frac{31}{9}=

We divide 52 52 by 4 4 and solve

13Γ—319= 13\times\frac{31}{9}=

4039=4479 \frac{403}{9}=44\frac{7}{9}

Answer:

4479 44\frac{7}{9}


Exercise 3

Task:

(7+2+3)(7+6)(12βˆ’3βˆ’4)=?(7+2+3)(7+6)(12-3-4)=\text{?}

Solution:

We start by solving each of the parentheses in the order of operations.

(9+3)Γ—13Γ—(9βˆ’4)= (9+3)\times13\times(9-4)=

12Γ—13Γ—5= 12\times13\times5=

We move the 5 to the left so that we can easily solve the exercise from left to right

12Γ—5Γ—13= 12\times5\times13=

60Γ—13=780 60\times13=780

Answer:

780 780


Do you know what the answer is?

Exercise 4

Task:

5β‹…7β‹…13β‹…6=?5\cdot7\cdot13\cdot6=\text{?}

Solution:

We reorder the exercise into two smaller expressions so that they are simpler to solve.

7β‹…13β‹…6⋅⁑5= 7\cdot13\cdot6\operatorname{\cdot}5=

We start by solving the first expression in the exercise, and then the second.

Then we solve the whole, simplified expression.

91β‹…30=2730 91\cdot30=2730

Answer:

2730 2730


Exercise 5

Task:

5β‹…17β‹…2=?5\cdot17\cdot2=\text{?}

Solution:

We reorder the exercise to make it easier to solve.

5β‹…2β‹…17= 5\cdot2\cdot17=

We continue solving the exercise from left to right.

10β‹…17=170 10\cdot17=170

Answer:

170 170


Check your understanding

Review questions

What is the commutative property in multiplication?

The commutative property of multiplication tells us that changing the order of the factors in an expression won't change the end result

For example, we will get the same answer if we multiply 3Γ—16 3\times16 or if we multiply 16Γ—3 16\times3 , in both cases the result is 48 48.


How does the commutative property work in multiplication?

The commutative property in multiplication can be understood as follows: aΓ—b=bΓ—a a\times b=b\times a

This means that the order of the factors does not change the result. If we assign values to a=4 a=4 and b=6 b=6 , applying the commutative property we will get:

4Γ—6=24 4\times6=24

6Γ—4=24 6\times4=24

We can see that in both cases it gave us the same result even though the factors are in a different order.


What are the 4 properties of multiplication?

The 4 properties of multiplication are the following:

  • Commutative: No matter the order of the factors, the result will be the same aΓ—b=bΓ—a a\times b=b\times a
  • Associative: No matter how we group sets of factors, the result will be the same. aΓ—(bΓ—c)=(aΓ—b)Γ—c a\times\left(b\times c\right)=\left(a\times b\right)\times c
  • Distributive: Multiplying a factor by the sum of two addends is the same as multiplying by each addend separately and then adding the results together. aΓ—(b+c)=aΓ—b+aΓ—c a\times\left(b+c\right)=a\times b+a\times c
  • Identity element: The identity element of multiplication is 1, because if we multiply a number by the number 1 1 , the result will still be that number. This can be understood as follows aΓ—1=a a\times1=a .

Do you think you will be able to solve it?

examples with solutions for the commutative property of multiplication

Exercise #1

Solve:

βˆ’5+4+1βˆ’3 -5+4+1-3

Video Solution

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

Answer

βˆ’3 -3

Exercise #2

4:2+2= 4:2+2=

Video Solution

Step-by-Step Solution

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

4:2=2 4:2=2

Now we obtain the exercise:

2+2=4 2+2=4

Answer

4 4

Exercise #3

14Γ—4+2= \frac{1}{4}\times4+2=

Video Solution

Step-by-Step Solution

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

We add the 4 in the numerator of the fraction:

1Γ—44+2= \frac{1\times4}{4}+2=

We solve the exercise in the numerator of the fraction and obtain:

44+2=1+2=3 \frac{4}{4}+2=1+2=3

Answer

3 3

Exercise #4

βˆ’2βˆ’4+6βˆ’1= -2-4+6-1=

Video Solution

Step-by-Step Solution

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

βˆ’2βˆ’4=βˆ’6 -2-4=-6

βˆ’6+6=0 -6+6=0

0βˆ’1=βˆ’1 0-1=-1

Answer

βˆ’1 -1

Exercise #5

12Γ—13+14= 12\times13+14=

Video Solution

Step-by-Step Solution

According to the order of operations, we start with the multiplication exercise and then with the addition.

12Γ—13=156 12\times13=156

Now we get the exercise:

156+14=170 156+14=170

Answer

170 170

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