Distributive Property: Rewriting 8(2n + 8)
Let's dive into understanding how to rewrite the expression 8(2n + 8) using the distributive property. This fundamental concept in mathematics allows us to simplify expressions by multiplying a term outside the parentheses with each term inside the parentheses. Mastering the distributive property is crucial for solving algebraic equations and simplifying complex expressions. So, let’s break down the process step by step and ensure you have a solid grasp of this essential mathematical tool.
Understanding the Distributive Property
The distributive property is a cornerstone of algebra, and it's essential to understand it thoroughly. At its heart, the distributive property is a mathematical rule that allows you to multiply a single term by two or more terms inside a set of parentheses. This property is especially useful when dealing with expressions that involve both addition and multiplication. The basic idea is that you 'distribute' the term outside the parentheses to each term inside the parentheses, ensuring that every term inside is multiplied by the term outside. This process helps in simplifying complex expressions and making them easier to work with.
For instance, consider the expression a(b + c). The distributive property tells us that this expression is equivalent to ab + ac. You are essentially multiplying 'a' by both 'b' and 'c' and then adding the results together. This simple yet powerful rule is the foundation for many algebraic manipulations and is critical in solving equations. Understanding how to apply the distributive property correctly can save you time and effort, and it is a skill that will be used repeatedly in higher-level mathematics.
Moreover, the distributive property is not just limited to simple addition within parentheses; it applies to subtraction as well. For example, a(b - c) is equivalent to ab - ac. The same principle applies: the term outside the parentheses is multiplied by each term inside, maintaining the operation (in this case, subtraction) between them. This versatility makes the distributive property a powerful tool in various mathematical contexts. Whether you're simplifying algebraic expressions, solving equations, or working with more complex mathematical problems, the distributive property remains a fundamental concept to master. Its consistent application ensures accurate and efficient problem-solving, paving the way for advanced mathematical understanding.
Applying the Distributive Property to 8(2n + 8)
When we apply the distributive property to the expression 8(2n + 8), we are essentially multiplying 8 by each term inside the parentheses individually. This means we need to multiply 8 by 2n and then multiply 8 by 8. Let’s break it down step by step to ensure clarity and understanding.
First, we multiply 8 by 2n. This can be written as 8 * (2n). When multiplying a number by a term that includes a variable, we multiply the coefficients (the numerical parts) together. In this case, we multiply 8 by 2, which gives us 16. So, 8 * (2n) simplifies to 16n. This is the first part of our distribution process, and it shows how the 8 interacts with the first term inside the parentheses.
Next, we multiply 8 by the second term inside the parentheses, which is 8. This is a straightforward multiplication: 8 * 8. The result of this multiplication is 64. This completes the distribution process, as we have now multiplied 8 by both terms inside the parentheses. It's crucial to ensure that you multiply the term outside the parentheses by every term inside to correctly apply the distributive property. By distributing the 8 across both 2n and 8, we maintain the equivalence of the expression while simplifying its form. This is the core idea behind the distributive property: rewriting expressions in a way that makes them easier to manipulate and solve.
By understanding this step-by-step process, you can confidently apply the distributive property to similar expressions. Remember, the key is to ensure each term inside the parentheses is multiplied by the term outside. This foundational skill is essential for algebraic problem-solving and will prove invaluable as you progress in mathematics. Careful and methodical application of the distributive property can help avoid common errors and simplify complex calculations.
Step-by-Step Breakdown: 8(2n + 8)
To effectively rewrite the expression 8(2n + 8) using the distributive property, we will follow a clear, step-by-step process. This breakdown will not only show the solution but also help you understand the mechanics behind the property, making it easier to apply in future scenarios. Each step is designed to build upon the previous one, ensuring a comprehensive grasp of the concept.
Step 1: Identify the Terms The first step is to identify the terms both outside and inside the parentheses. In the expression 8(2n + 8), the term outside the parentheses is 8, and the terms inside are 2n and 8. Recognizing these components is crucial for the correct application of the distributive property. It sets the stage for the subsequent multiplication steps.
Step 2: Distribute the Outer Term to the First Inner Term Next, we multiply the outer term (8) by the first term inside the parentheses (2n). This means we perform the operation 8 * 2n. As explained earlier, we multiply the coefficients (8 and 2) to get 16, resulting in the term 16n. This step effectively distributes the 8 to the first part of the expression inside the parentheses.
Step 3: Distribute the Outer Term to the Second Inner Term Now, we multiply the outer term (8) by the second term inside the parentheses (8). This is a straightforward multiplication: 8 * 8. The result is 64. This step completes the distribution process by accounting for the second term inside the parentheses.
Step 4: Combine the Results Finally, we combine the results from Step 2 and Step 3. We have 16n from the first distribution and 64 from the second. We add these two terms together to form our equivalent expression. This gives us 16n + 64. This final expression represents the simplified form of the original expression after applying the distributive property.
By following these steps, we have successfully rewritten 8(2n + 8) as 16n + 64. Each step is deliberate and crucial, ensuring that the distributive property is applied correctly. This step-by-step approach not only provides the answer but also enhances your understanding of how the distributive property works, making it easier to apply in more complex situations. Regular practice with this method will solidify your skills and confidence in algebraic manipulations.
The Equivalent Expression
After meticulously applying the distributive property to the expression 8(2n + 8), we arrive at the equivalent expression: 16n + 64. This result is crucial because it simplifies the original expression while maintaining its mathematical value. Understanding how we reached this equivalent form is vital for grasping the practical application of the distributive property.
The process began with multiplying 8 by each term inside the parentheses. First, we multiplied 8 by 2n, resulting in 16n. This step demonstrated how the term outside the parentheses interacts with the variable term inside. The multiplication of the coefficients (8 and 2) gave us the new coefficient for the variable n, illustrating a key aspect of algebraic manipulation. Next, we multiplied 8 by 8, which yielded 64. This step completed the distribution by accounting for the constant term inside the parentheses.
Combining these results, we formed the expression 16n + 64. This equivalent expression is simpler and easier to work with in many contexts. For example, if we were solving an equation, having the expression in this form would allow us to isolate the variable and find its value more efficiently. The distributive property, therefore, serves as a powerful tool for transforming expressions into more manageable forms.
The expression 16n + 64 accurately represents the initial expression 8(2n + 8) after the distributive property has been correctly applied. This final result underscores the importance of understanding and applying the distributive property to simplify algebraic expressions. It showcases how a seemingly complex expression can be transformed into a more straightforward one, facilitating further mathematical operations and problem-solving.
Why This Matters: Real-World Applications
The distributive property isn't just an abstract mathematical concept; it has practical applications in various real-world scenarios. Understanding this property can help simplify problems in everyday situations, from calculating costs to planning projects. Recognizing these real-world applications can also make learning mathematics more engaging and relevant.
One common application is in calculating expenses. Imagine you're buying 8 identical sets of items, each costing 2n dollars plus an additional 8 dollars for shipping. This situation is directly analogous to the expression 8(2n + 8). Using the distributive property, you can easily break down the total cost. Multiplying 8 by 2n gives you the total cost of the items (16n dollars), and multiplying 8 by 8 gives you the total shipping cost (64 dollars). Adding these together (16n + 64 dollars) provides the total expense, making it easier to budget and plan your purchase.
Another application of the distributive property is in geometry, particularly when calculating areas. Suppose you have a rectangular garden with a width of 8 feet and a length that is 2n feet plus an additional 8 feet. The area of the garden can be calculated using the formula Area = width * length, which in this case is 8(2n + 8). By distributing the 8, you find that the area is 16n + 64 square feet. This can be useful for determining the amount of fencing needed or the quantity of soil to purchase.
Project planning also benefits from the distributive property. Consider a scenario where you're organizing an event with 8 teams, and each team needs 2n resources plus 8 additional items. The total resources needed can be represented by the expression 8(2n + 8). Applying the distributive property allows you to quickly calculate the total number of resources required (16n) and the total number of additional items (64). This helps in efficient planning and resource allocation.
These examples highlight the versatility of the distributive property and its relevance in everyday problem-solving. By mastering this concept, you equip yourself with a powerful tool that extends beyond the classroom, enabling you to tackle a wide range of practical challenges. Recognizing these real-world connections can make mathematical learning more meaningful and applicable.
In conclusion, we have successfully used the distributive property to rewrite the expression 8(2n + 8) as 16n + 64. This process not only simplifies the expression but also demonstrates the practical application of a fundamental mathematical concept. Remember, the key to mastering the distributive property is to practice and apply it in various contexts. For further exploration of algebraic properties and their applications, you might find helpful resources on websites like Khan Academy Algebra. Keep practicing, and you'll become more confident in your algebraic skills!
