Understanding The Distributive Law Which Statement Applies It Correctly

The distributive law is a fundamental concept in mathematics, particularly in algebra, that describes how multiplication interacts with addition and subtraction. It's a cornerstone for simplifying expressions and solving equations, making it crucial for students and anyone working with mathematical operations. Understanding and applying the distributive law correctly is essential for success in more advanced mathematical topics. In this article, we'll delve into the distributive law, explore its applications, and identify which statement accurately demonstrates its use. We will break down the concept with clear explanations and examples, ensuring you grasp this vital mathematical principle.

Defining the Distributive Law

At its core, the distributive law states that multiplying a single term by a sum or difference inside parentheses is the same as multiplying the term by each part of the sum or difference individually and then performing the addition or subtraction. This seemingly simple concept is a powerful tool for expanding expressions and simplifying equations. The distributive property allows us to break down complex expressions into manageable parts. It connects multiplication and addition (or subtraction) in a unique and useful way. To fully grasp the concept, let's delve deeper into the mathematical representation and explore some illustrative examples.

The Mathematical Representation

The distributive law can be represented algebraically as follows:

• a × (b + c) = (a × b) + (a × c)
• a × (b - c) = (a × b) - (a × c)

Where 'a', 'b', and 'c' can be any real numbers. The variable 'a' is being distributed over the terms 'b' and 'c'. This means that 'a' is multiplied by both 'b' and 'c' individually, and the resulting products are either added or subtracted, depending on the original operation within the parentheses. This algebraic representation is the formal way of expressing the distributive law, and it is essential for solving a wide range of mathematical problems. The key takeaway here is that distribution involves breaking down a multiplication operation into multiple smaller operations, making the overall calculation more manageable.

Illustrative Examples

To illustrate the distributive law in action, let's consider a few examples:

1. Example 1: 3 × (4 + 2)

   Using the distributive law:

   3 × (4 + 2) = (3 × 4) + (3 × 2) = 12 + 6 = 18

   Direct calculation: 3 × (6) = 18

2. Example 2: 5 × (7 - 3)

   Using the distributive law:

   5 × (7 - 3) = (5 × 7) - (5 × 3) = 35 - 15 = 20

   Direct calculation: 5 × (4) = 20

These examples demonstrate how the distributive law works in practice. By distributing the term outside the parentheses, we can simplify the expression and arrive at the correct answer. The distributive law not only simplifies calculations but also provides a flexible approach to solving problems. It allows us to transform a single multiplication operation into a series of simpler operations, which can be particularly useful when dealing with larger numbers or algebraic expressions.

Analyzing the Given Statements

Now, let's analyze the given statements to identify which one correctly applies the distributive law:

$\begin{array}{l} 9 \times(5+3)=9 \times(3+5) \\ 9 \times(5+3)=9 \times 5+9 \times 3 \\ 9 \times(5+3)=(9 \times 5)+3 \\ 9 \times(5+3)=9 \times 8 \end{array}$

We will examine each statement step by step to determine if it follows the principles of the distributive law. Understanding why a statement is correct or incorrect is crucial for mastering the distributive property and applying it effectively in various mathematical contexts. Let's break down each option and see which one aligns perfectly with the distributive law.

Statement 1: 9 × (5 + 3) = 9 × (3 + 5)

This statement demonstrates the commutative property of addition within the parentheses. The commutative property states that changing the order of addends does not change the sum (a + b = b + a). While this statement is mathematically correct, it does not illustrate the distributive law. The distributive law involves multiplying a term across a sum or difference, which is not what's happening here. The commutative property is a fundamental concept in arithmetic, but it's distinct from the distributive property. Understanding the difference between these properties is crucial for accurate mathematical reasoning.

Statement 2: 9 × (5 + 3) = 9 × 5 + 9 × 3

This statement correctly applies the distributive law. It shows that multiplying 9 by the sum of 5 and 3 is the same as multiplying 9 by 5 and 9 by 3 separately, and then adding the products. This aligns perfectly with the definition of the distributive law: a × (b + c) = (a × b) + (a × c). This statement is a clear and accurate example of how the distributive law works in practice. It effectively breaks down the multiplication operation into smaller, more manageable parts, which is the essence of distribution.

Statement 3: 9 × (5 + 3) = (9 × 5) + 3

This statement incorrectly applies the distributive law. It only multiplies 9 by 5 and then adds 3, neglecting to multiply 9 by the second term inside the parentheses. The distributive law requires multiplying the term outside the parentheses by each term inside. This statement demonstrates a misunderstanding of the distributive property, as it fails to distribute the multiplication across all terms within the parentheses. The result is an incorrect application of the distributive law, leading to an inaccurate simplification of the expression.

Statement 4: 9 × (5 + 3) = 9 × 8

This statement simply evaluates the expression inside the parentheses first, following the order of operations (PEMDAS/BODMAS). While mathematically correct, it does not demonstrate the distributive law. It's a straightforward calculation that simplifies the expression but doesn't involve the distribution of multiplication over addition. This statement showcases the importance of following the order of operations, but it does not highlight the specific application of the distributive law. Therefore, while the arithmetic is accurate, it's not relevant to the concept we're exploring.

Conclusion

Therefore, the statement that uses the distributive law is:

9 × (5 + 3) = 9 × 5 + 9 × 3

Understanding the distributive law is crucial for simplifying expressions and solving equations in algebra and beyond. This property allows us to break down complex multiplication problems into simpler steps, making them easier to manage. Mastering the distributive law is not just about memorizing a formula; it's about understanding the underlying principle of how multiplication interacts with addition and subtraction. By recognizing and applying the distributive law correctly, you'll be well-equipped to tackle a wide range of mathematical challenges. This fundamental concept is a building block for more advanced mathematical topics, making it an essential skill for anyone pursuing further studies in mathematics or related fields.