Which Of The Following Expressions Demonstrates The Distributive Property?A. $3 + 4 + 5 = 4 + 3 + 5$B. $-2(5 + 7) = -2(7 + 5$\]C. $3(-8 + 1) = 3(-8) + 3(1$\]D. $6[(7)(-2)] = [(6)(7)](-2$\]

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The distributive property is a fundamental concept in mathematics that allows us to expand expressions by multiplying each term inside the parentheses with the factor outside. It is a crucial property that helps us simplify complex expressions and solve equations. In this article, we will explore the distributive property and identify which of the given expressions demonstrates this property.

What is the Distributive Property?

The distributive property states that for any real numbers a, b, and c, the following equation holds:

a(b + c) = ab + ac

This means that we can distribute the factor a to each term inside the parentheses, b and c, and then combine the results.

Examples of the Distributive Property

Let's consider some examples to illustrate the distributive property:

  • 2(3 + 4) = 2(3) + 2(4) = 6 + 8 = 14
  • 5(x + 2) = 5x + 10
  • 3(2y - 4) = 6y - 12

As we can see, the distributive property allows us to expand expressions by multiplying each term inside the parentheses with the factor outside.

Which Expression Demonstrates the Distributive Property?

Now, let's examine the given expressions and determine which one demonstrates the distributive property:

A. 3+4+5=4+3+53 + 4 + 5 = 4 + 3 + 5

This expression does not demonstrate the distributive property. It is simply a commutative property, which states that the order of the terms does not change the result.

B. −2(5+7)=−2(7+5)-2(5 + 7) = -2(7 + 5)

This expression does not demonstrate the distributive property. It is simply a commutative property, which states that the order of the terms does not change the result.

C. 3(−8+1)=3(−8)+3(1)3(-8 + 1) = 3(-8) + 3(1)

This expression demonstrates the distributive property. We can see that the factor 3 is distributed to each term inside the parentheses, -8 and 1, and then combined the results.

D. 6[(7)(−2)]=[(6)(7)](−2)6[(7)(-2)] = [(6)(7)](-2)

This expression does not demonstrate the distributive property. It is simply a commutative property, which states that the order of the terms does not change the result.

Conclusion

In conclusion, the expression 3(−8+1)=3(−8)+3(1)3(-8 + 1) = 3(-8) + 3(1) demonstrates the distributive property. It shows that the factor 3 is distributed to each term inside the parentheses, -8 and 1, and then combined the results. This is a fundamental concept in mathematics that helps us simplify complex expressions and solve equations.

Real-World Applications of the Distributive Property

The distributive property has numerous real-world applications in various fields, including:

  • Algebra: The distributive property is used to simplify complex expressions and solve equations.
  • Geometry: The distributive property is used to find the area and perimeter of shapes.
  • Physics: The distributive property is used to describe the motion of objects and the forces acting on them.
  • Engineering: The distributive property is used to design and optimize systems.

Tips and Tricks for Applying the Distributive Property

Here are some tips and tricks for applying the distributive property:

  • Always distribute the factor to each term inside the parentheses.
  • Combine the results of the distribution.
  • Use the distributive property to simplify complex expressions.
  • Use the distributive property to solve equations.

Common Mistakes to Avoid When Applying the Distributive Property

Here are some common mistakes to avoid when applying the distributive property:

  • Not distributing the factor to each term inside the parentheses.
  • Not combining the results of the distribution.
  • Using the distributive property incorrectly.
  • Not using the distributive property when it is applicable.

Conclusion

The distributive property is a fundamental concept in mathematics that can be confusing for many students. In this article, we will answer some frequently asked questions about the distributive property to help you better understand this concept.

Q: What is the distributive property?

A: The distributive property is a mathematical property that allows us to expand expressions by multiplying each term inside the parentheses with the factor outside. It states that for any real numbers a, b, and c, the following equation holds:

a(b + c) = ab + ac

Q: How do I apply the distributive property?

A: To apply the distributive property, you need to follow these steps:

  1. Identify the factor outside the parentheses.
  2. Multiply the factor by each term inside the parentheses.
  3. Combine the results of the multiplication.

Q: What are some examples of the distributive property?

A: Here are some examples of the distributive property:

  • 2(3 + 4) = 2(3) + 2(4) = 6 + 8 = 14
  • 5(x + 2) = 5x + 10
  • 3(2y - 4) = 6y - 12

Q: Can I use the distributive property with negative numbers?

A: Yes, you can use the distributive property with negative numbers. For example:

  • -2(3 + 4) = -2(3) + -2(4) = -6 - 8 = -14
  • -3(x - 2) = -3x + 6

Q: Can I use the distributive property with fractions?

A: Yes, you can use the distributive property with fractions. For example:

  • 1/2(3 + 4) = 1/2(3) + 1/2(4) = 3/2 + 2 = 7/2
  • 3/4(x + 2) = 3/4x + 3/2

Q: Can I use the distributive property with decimals?

A: Yes, you can use the distributive property with decimals. For example:

  • 2.5(3 + 4) = 2.5(3) + 2.5(4) = 7.5 + 10 = 17.5
  • 0.5(x - 2) = 0.5x - 1

Q: Can I use the distributive property with variables?

A: Yes, you can use the distributive property with variables. For example:

  • 2(x + 3) = 2x + 6
  • 3(y - 2) = 3y - 6

Q: What are some common mistakes to avoid when applying the distributive property?

A: Here are some common mistakes to avoid when applying the distributive property:

  • Not distributing the factor to each term inside the parentheses.
  • Not combining the results of the distribution.
  • Using the distributive property incorrectly.
  • Not using the distributive property when it is applicable.

Q: How can I practice the distributive property?

A: Here are some ways to practice the distributive property:

  • Use online resources such as Khan Academy or Mathway to practice the distributive property.
  • Work with a tutor or teacher to practice the distributive property.
  • Use worksheets or practice problems to practice the distributive property.
  • Apply the distributive property to real-world problems.

Conclusion

In conclusion, the distributive property is a fundamental concept in mathematics that can be confusing for many students. By understanding the distributive property and applying it correctly, you can solve a wide range of mathematical problems and make informed decisions in various fields. Remember to practice the distributive property regularly to become proficient in this concept.