The Commutative Property of Multiplication is one of the four main properties of multiplication. It is named after the ability of factors to commute, or move, in the number sentence without affecting the product.

**Commutative Property of Multiplication** says that the order of factors in a multiplication sentence has no effect on the product. The Commutative Property of Multiplication works on integers, fractions, decimals, exponents, and algebraic equations.

The word “commutative” comes from a Latin root meaning “interchangeable”.

Switching the order of the multiplicand (the first factor) and the multiplier (the second factor) does not change the product.

What is $4\times 5$? The answer is $20$.

What is $5\times 4$? The answer is also $20$.

The order of the two factors, $4$ and $5$, did not affect the product, $20$.

Commutative property is also true for addition. Learn about the commutative property of addition.

The **generic formula** for the Commutative Property of Multiplication is:

$ab=ba$

Any number of factors can be rearranged to yield the same product:

- $1\times 2\times 3=3\times 1\times 2=\mathbf{6}=2\times 3\times 1=2\times 1\times 3$

Often, when demonstrating the commutative property of multiplication, the product is shown in the middle of the multiple arrangements of the equation.

Let's see these fancy words in action:

*[insert drawing of an ordered array of four rows of five rubber ducks; to the right is a second drawing of five rows of four rubber ducks]*

In the first picture we can think of the set of five rubber ducks as the multiplicand, spread across from left to right. Beneath it, vertically, we have the multiplier, $4$.

In the second picture we have one set of four rubber ducks arrayed left to right, the multiplicand. Then we have the multiplier, $5$, vertically.

Whether we take a set of five rubber duckies and multiply them four times, as on the left, or we take a set of four rubber duckies and multiply them five times, as on the right, we still end up with $20$ rubber duckies.

The Commutative Property of Multiplication works on basic multiplication equations and algebraic equations. Here was see how to use commutative property of multiplication various multiplication sentences:

**Integers:**

- $6\times 7=\mathbf{42}=7\times 6$
- $\mathrm{1,234}*0=\mathbf{0}=0*\mathrm{1,234}$
- $717\times 11=\mathbf{7,887}=11\times 717$

**Exponents:**

- ${6}^{2}\times {3}^{2}=\mathbf{324}={3}^{2}\times {6}^{2}$
- ${2}^{3}\times {4}^{3}=\mathbf{256}={4}^{3}\times {2}^{3}$

**Fractions:**

- $\frac{3}{4}\times \frac{7}{8}=\frac{\mathbf{21}}{\mathbf{32}}=\left(\frac{7}{8}\right)\left(\frac{3}{4}\right)$
- $\frac{9}{10}\times \frac{75}{100}=\frac{75}{100}\times \frac{9}{10}=\frac{\mathbf{675}}{\mathbf{1000}}\mathbf{=}\frac{\mathbf{27}}{\mathbf{40}}$ (simplified by dividing by $\frac{25}{25}$)

**Decimals:**

- $0.1234\times 0.987=\mathbf{0.1217958}=0.987\times 0.1234$
- $411.52\times 0.3=\mathbf{123.456}=0.3\times 411.52$

**Variables:**

- $4{x}^{2}\left(2\right)=\mathbf{32}=2\left(4{x}^{2}\right)$

To get our answer $32$, we first solved for $x$.

After working your way through this lesson and video, you have learned:

- Defintion of commutative property of multiplication means
- Commutative property of multiplication formula
- How to use the commutative property of multiplication
- Examples of cummutative property of multiplication

Instructor: **Malcolm M.**

Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

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