Demonstrating the Commutative Property in Multiplication

Oct 13, 2025 by miodeya.com 86 views

The commutative property in mathematics is a fundamental concept that allows us to rearrange the order of numbers in an equation without changing the result. In the context of multiplication, the commutative property states that the order of the numbers being multiplied does not affect the product. In this article, we will explore the commutative property across multiplication and determine which of the given options demonstrates this property for the expression $2 \times 10 \times 7$ .

What is the Commutative Property?

The commutative property is a mathematical concept that applies to both addition and multiplication. In the case of addition, the commutative property states that the order of the numbers being added does not affect the sum. For example, $a + b = b + a$ . Similarly, in the case of multiplication, the commutative property states that the order of the numbers being multiplied does not affect the product. For example, $a \times b = b \times a$ .

Demonstrating the Commutative Property in Multiplication

To demonstrate the commutative property in multiplication, we need to show that the order of the numbers being multiplied does not affect the product. Let's consider the expression $2 \times 10 \times 7$ . We can rearrange the order of the numbers in this expression to demonstrate the commutative property.

Option A: $2 + 10 \times 7$

Option A is not a demonstration of the commutative property in multiplication. The expression $2 + 10 \times 7$ involves addition, not multiplication. Therefore, this option does not demonstrate the commutative property in multiplication.

Option B: $10 \times 7 \times 2$

Option B is a demonstration of the commutative property in multiplication. The expression $10 \times 7 \times 2$ is equivalent to the original expression $2 \times 10 \times 7$ . The order of the numbers being multiplied has been rearranged, but the product remains the same. This demonstrates the commutative property in multiplication.

Option C: $(2 \times 10) \times 7$

Option C is not a demonstration of the commutative property in multiplication. The expression $(2 \times 10) \times 7$ involves the evaluation of the expression inside the parentheses first, which is $20$ . Then, the result is multiplied by $7$ , resulting in $140$ . This is not equivalent to the original expression $2 \times 10 \times 7$ , which equals $140$ . However, this option does not demonstrate the commutative property in multiplication because it involves the evaluation of the expression inside the parentheses first.

Option D: $2 \times (10 \times 7)$

Option D is not a demonstration of the commutative property in multiplication. The expression $2 \times (10 \times 7)$ involves the evaluation of the expression inside the parentheses first, which is $70$ . Then, the result is multiplied by $2$ , resulting in $140$ . This is not equivalent to the original expression $2 \times 10 \times 7$ , which equals $140$ . However, this option does not demonstrate the commutative property in multiplication because it involves the evaluation of the expression inside the parentheses first.

Conclusion

In conclusion, the commutative property in multiplication states that the order of the numbers being multiplied does not affect the product. To demonstrate this property, we need to show that the order of the numbers being multiplied does not affect the product. Option B, $10 \times 7 \times 2$ , is a demonstration of the commutative property in multiplication because it is equivalent to the original expression $2 \times 10 \times 7$ . The order of the numbers being multiplied has been rearranged, but the product remains the same.

Understanding the Commutative Property in Real-World Applications

The commutative property in multiplication has many real-world applications. For example, in finance, the commutative property can be used to calculate the total cost of a product when the prices of the individual components are known. In engineering, the commutative property can be used to calculate the total force exerted on an object when the forces exerted by individual components are known.

Real-World Example: Calculating the Total Cost of a Product

Suppose we want to calculate the total cost of a product that consists of three components: a CPU, a motherboard, and a hard drive. The prices of the individual components are $100, $200, and $300, respectively. We can use the commutative property in multiplication to calculate the total cost of the product.

Let's consider the expression $100 \times 200 \times 300$ . We can rearrange the order of the numbers being multiplied to demonstrate the commutative property. For example, we can multiply the first two numbers first, resulting in $20,000$ . Then, we can multiply the result by the third number, resulting in $6,000,000$ . This is equivalent to the original expression $100 \times 200 \times 300$ , which equals $6,000,000$ .

Conclusion

In conclusion, the commutative property in multiplication is a fundamental concept that allows us to rearrange the order of numbers in an equation without changing the result. Option B, $10 \times 7 \times 2$ , is a demonstration of the commutative property in multiplication because it is equivalent to the original expression $2 \times 10 \times 7$ . The commutative property has many real-world applications, including finance and engineering. By understanding the commutative property in multiplication, we can calculate the total cost of a product and the total force exerted on an object.

Final Thoughts

The commutative property in multiplication is a fundamental concept in mathematics that allows us to rearrange the order of numbers in an equation without changing the result. In this article, we will answer some frequently asked questions about the commutative property in multiplication.

Q: What is the commutative property in multiplication?

A: The commutative property in multiplication is a mathematical concept that states that the order of the numbers being multiplied does not affect the product. In other words, the commutative property allows us to rearrange the order of numbers in an equation without changing the result.

Q: How does the commutative property in multiplication work?

A: The commutative property in multiplication works by allowing us to rearrange the order of numbers in an equation without changing the result. For example, if we have the equation $2 \times 10 \times 7$ , we can rearrange the order of the numbers to get $10 \times 7 \times 2$ . The result of both equations is the same, which is $140$ .

Q: What are some real-world applications of the commutative property in multiplication?

A: The commutative property in multiplication has many real-world applications, including finance and engineering. For example, in finance, the commutative property can be used to calculate the total cost of a product when the prices of the individual components are known. In engineering, the commutative property can be used to calculate the total force exerted on an object when the forces exerted by individual components are known.

Q: Can the commutative property in multiplication be used with addition?

A: No, the commutative property in multiplication cannot be used with addition. The commutative property only applies to multiplication, not addition. However, there is a similar concept called the commutative property of addition, which states that the order of the numbers being added does not affect the sum.

Q: What are some common mistakes to avoid when using the commutative property in multiplication?

A: Some common mistakes to avoid when using the commutative property in multiplication include:

Not following the order of operations (PEMDAS)
Not evaluating expressions inside parentheses first
Not using the correct order of numbers in an equation
Not checking the result of an equation to ensure it is correct

Q: How can I practice using the commutative property in multiplication?

A: There are many ways to practice using the commutative property in multiplication, including:

Using online math resources and games
Practicing with worksheets and exercises
Working with a tutor or teacher
Using real-world examples and applications

Q: What are some advanced concepts related to the commutative property in multiplication?

A: Some advanced concepts related to the commutative property in multiplication include:

The associative property of multiplication
The distributive property of multiplication
The commutative property of addition
The associative property of addition

Conclusion

In conclusion, the commutative property in multiplication is a fundamental concept in mathematics that allows us to rearrange the order of numbers in an equation without changing the result. By understanding the commutative property, we can simplify complex calculations and solve problems more efficiently. We hope this article has helped to answer some frequently asked questions about the commutative property in multiplication.

Additional Resources

For more information on the commutative property in multiplication, please see the following resources:

Khan Academy: Commutative Property of Multiplication
Math Is Fun: Commutative Property of Multiplication
Wolfram MathWorld: Commutative Property of Multiplication

Final Thoughts

The commutative property in multiplication is a powerful tool that can be used to simplify complex calculations. By understanding the commutative property, we can rearrange the order of numbers in an equation without changing the result. We hope this article has helped to answer some frequently asked questions about the commutative property in multiplication.