Distributive Property: Multiplication Over Addition
The distributive property is a fundamental concept in mathematics that allows us to simplify expressions involving multiplication and addition. It states that multiplying a number by the sum of two other numbers is the same as multiplying the number by each of the other numbers individually and then adding the products. In simpler terms, it lets you 'distribute' a factor across terms within parentheses. Understanding and applying the distributive property is crucial for solving algebraic equations and simplifying complex expressions. Let's delve deeper into what the distributive property entails and why it's such a valuable tool in mathematics.
Understanding the Distributive Property
The distributive property can be expressed as:
- a × (b + c) = (a × b) + (a × c)
Where 'a', 'b', and 'c' represent any real numbers. This formula tells us that if we have a number 'a' multiplied by the sum of 'b' and 'c', we can distribute 'a' to both 'b' and 'c' individually, multiply them, and then add the results. This property is incredibly useful because it allows us to break down complex expressions into simpler, more manageable parts. For example, consider the expression 3 × (2 + 4). According to the distributive property, this is equivalent to (3 × 2) + (3 × 4), which simplifies to 6 + 12, and finally equals 18. This simple example illustrates the power of the distributive property in simplifying arithmetic and algebraic expressions.
Why is the Distributive Property Important?
The distributive property is not just a mathematical curiosity; it is a cornerstone of algebra and arithmetic. Its importance stems from several key benefits:
- Simplifying Expressions: The distributive property allows us to simplify complex expressions, making them easier to understand and solve. By distributing a factor across terms within parentheses, we can break down an expression into smaller, more manageable parts.
- Solving Equations: In algebra, the distributive property is essential for solving equations. It enables us to eliminate parentheses and combine like terms, which are crucial steps in isolating variables and finding solutions.
- Performing Mental Calculations: The distributive property can also be used to perform mental calculations more easily. For example, to multiply 7 by 102, you can think of 102 as (100 + 2), and then distribute the 7: 7 × (100 + 2) = (7 × 100) + (7 × 2) = 700 + 14 = 714.
- General Mathematical Understanding: Understanding the distributive property helps build a strong foundation for more advanced mathematical concepts. It reinforces the relationship between multiplication and addition, and it provides a framework for understanding more complex algebraic manipulations.
Analyzing the Given Options
Now, let's analyze the given options to identify the one that correctly illustrates the distributive property of multiplication over addition:
A. 4 × (9 + 7) = (4 + 9) × (4 + 7) B. 4 × (9 + 7) = (4 × 9) + 7 C. 4 × (9 + 7) = (4 × 9) + (4 × 7)
To determine the correct option, we need to compare each one to the general form of the distributive property: a × (b + c) = (a × b) + (a × c).
Option A: 4 × (9 + 7) = (4 + 9) × (4 + 7)
This option is incorrect because it does not follow the distributive property. Instead, it seems to be arbitrarily adding numbers within the parentheses and then multiplying them. The left side of the equation is 4 multiplied by the sum of 9 and 7, while the right side is the sum of 4 and 9 multiplied by the sum of 4 and 7. This is not a valid application of the distributive property.
Option B: 4 × (9 + 7) = (4 × 9) + 7
This option is also incorrect. It only multiplies 4 by 9 and then adds 7, but it does not distribute the multiplication over both terms inside the parentheses. According to the distributive property, 4 should be multiplied by both 9 and 7, and then the products should be added together. This option misses the crucial step of multiplying 4 by 7.
Option C: 4 × (9 + 7) = (4 × 9) + (4 × 7)
This option correctly illustrates the distributive property. It shows that 4 multiplied by the sum of 9 and 7 is equal to 4 multiplied by 9 plus 4 multiplied by 7. This follows the pattern a × (b + c) = (a × b) + (a × c), where a = 4, b = 9, and c = 7. Therefore, this is the correct application of the distributive property.
Conclusion
In conclusion, the correct expression that illustrates the distributive property of multiplication over addition is:
C. 4 × (9 + 7) = (4 × 9) + (4 × 7)
Understanding the distributive property is essential for simplifying expressions and solving equations in mathematics. It allows us to break down complex problems into smaller, more manageable parts, making it a valuable tool for both arithmetic and algebra.
By correctly applying the distributive property, we can simplify expressions, solve equations, and gain a deeper understanding of mathematical relationships. This fundamental concept is a building block for more advanced topics in mathematics, making it an essential skill for students and anyone working with mathematical concepts. Remember to distribute the factor correctly over each term within the parentheses, and you'll be well on your way to mastering this important property.
For further reading and a more in-depth understanding of the distributive property, you can visit Khan Academy's article on the Distributive Property.
