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Properties of Multiplication: Definition and Example | EDU.COMEDU.COMResourcesBlogGuidePodcastPlanBackHomesvg]:size-3.5">Math Glossarysvg]:size-3.5">Properties of MultiplicationProperties of Multiplication: Definition and ExampleTable of ContentsDefinition of Properties of Multiplication

Multiplication properties are specific rules or formulas that help simplify expressions involving multiplication. Since multiplication is defined as repeated addition (for example, 12×612 \times 612×6 means adding 121212 to itself 666 times, which equals 727272), these properties provide systematic ways to work with multiplicative expressions more efficiently. Understanding these properties enables us to solve complex mathematical problems with greater ease and flexibility.

The five fundamental properties of multiplication are: commutative, associative, distributive, identity, and zero property. Additional important properties include closure property, multiplication property of equality, and inverse property. The commutative property states that changing the order of factors doesn't affect the product. The associative property allows grouping numbers differently without changing the result. The distributive property connects multiplication with addition and subtraction. The identity property involves multiplication by 111, while the zero property relates to multiplication by 000.

Examples of Properties of Multiplication Example 1: Identifying Multiplication Properties Problem:

Identify the properties of multiplication used in each equation:

7×5=5×77 \times 5 = 5 \times 77×5=5×7 4×(3×8)=(4×3)×84 \times (3 \times 8) = (4 \times 3) \times 84×(3×8)=(4×3)×8 1×46=461 \times 46 = 461×46=46 34×134=134 \times \frac{1}{34} = 134×341​=1 Step-by-step solution: Step 1, first equation analysis: Notice that in 7×5=5×77 \times 5 = 5 \times 77×5=5×7, the order of the factors is changed. When we exchange the positions of numbers being multiplied and the result stays the same, we're using the commutative property. Therefore, 7×5=5×77 \times 5 = 5 \times 77×5=5×7 demonstrates the commutative property of multiplication. Step 2, second equation analysis: In 4×(3×8)=(4×3)×84 \times (3 \times 8) = (4 \times 3) \times 84×(3×8)=(4×3)×8, we're changing how the numbers are grouped. When we rearrange the grouping of factors without changing the result, we're using the associative property. Therefore, this equation demonstrates the associative property of multiplication. Step 3, third equation analysis: Looking at 1×46=461 \times 46 = 461×46=46, we see that multiplying by 111 gives the number itself. When multiplication by 111 yields the original number, we're using the identity property. Therefore, this equation demonstrates the identity property of multiplication. Step 4, fourth equation analysis: In 34×134=134 \times \frac{1}{34} = 134×341​=1, we're multiplying a number by its reciprocal. When a number is multiplied by its reciprocal and equals 111, we're using the inverse property. Therefore, this equation demonstrates the inverse property of multiplication. Example 2: Using the Distributive Property Problem:

Find the missing numbers in 12×(4+3)=‾+‾12 \times (4 + 3) = \underline{} + \underline{}12×(4+3)=​+​

Step-by-step solution: Step 1, identify the relevant property. This problem involves multiplication with a sum inside parentheses. We can apply the distributive property of multiplication over addition, which states that a(b+c)=ab+aca(b + c) = ab + aca(b+c)=ab+ac Step 2, apply the distributive property. In our problem, a=12a = 12a=12, b=4b = 4b=4, and c=3c = 3c=3 Using the formula a(b+c)=ab+aca(b + c) = ab + aca(b+c)=ab+ac, we get: 12×(4+3)=(12×4)+(12×3)12 \times (4 + 3) = (12 \times 4) + (12 \times 3)12×(4+3)=(12×4)+(12×3) Step 3, calculate each part. First part: 12×4=4812 \times 4 = 4812×4=48 Second part: 12×3=3612 \times 3 = 3612×3=36 Step 4, complete the equation. 12×(4+3)=48+3612 \times (4 + 3) = 48 + 3612×(4+3)=48+36 Therefore, the missing numbers are 484848 and 363636 Example 3: Using Multiple Properties Problem:

Find the product of 75×(−31)×17575 \times (-31) \times \frac{1}{75}75×(−31)×751​ using suitable properties.

Step-by-step solution: Step 1, identify helpful properties. We can use the commutative and associative properties to rearrange the factors. We can also use the inverse property for multiplication when we see a number and its reciprocal. Step 2, rearrange using commutative and associative properties. Original expression: 75×(−31)×17575 \times (-31) \times \frac{1}{75}75×(−31)×751​ Using commutative property to rearrange: 75×175×(−31)75 \times \frac{1}{75} \times (-31)75×751​×(−31) Using associative property to group: (75×175)×(−31)(75 \times \frac{1}{75}) \times (-31)(75×751​)×(−31) Step 3, apply the inverse property. We know that 75×175=175 \times \frac{1}{75} = 175×751​=1 (inverse property of multiplication) So our expression becomes: 1×(−31)1 \times (-31)1×(−31) Step 4, apply the identity property. Using the identity property, 1×(−31)=−311 \times (-31) = -311×(−31)=−31 Step 5, state the final answer. Therefore, 75×(−31)×175=−3175 \times (-31) \times \frac{1}{75} = -3175×(−31)×751​=−31 Comments(8)HHistoryTutorEthanNovember 6, 2025I've been using this glossary to help my students with multiplication properties. It's made explaining these concepts so much easier!

KKitesurferBobNovember 4, 2025I've used this glossary to teach multiplication properties. The clear defs and examples made it easy for my students to grasp. Great resource!

NNatureLover85September 17, 2025This glossary was a lifesaver for my kid! The examples made the properties of multiplication so easy to understand. We used the distributive property to simplify homework, and it clicked right away!

MCMs. CarterSeptember 10, 2025I’ve used the Properties of Multiplication examples on this page to help my kids with homework—they finally get the distributive property! Clear explanations and examples really made a difference.

MCMs. CarterAugust 27, 2025I used this page to help my kids understand the properties of multiplication, and it’s been a game changer! The examples are clear, and they’ve made homework so much easier. Highly recommend for parents!

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