Simplifying Expressions Using The Distributive Property 9(8x - 5)

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In the realm of mathematics, simplifying expressions is a fundamental skill that lays the groundwork for more advanced concepts. One of the key tools in our simplification arsenal is the distributive property. This property allows us to efficiently handle expressions involving parentheses and multiplication. In this comprehensive guide, we'll delve into the distributive property and apply it to simplify the expression 9(8x - 5) completely. We will meticulously walk through each step, ensuring a clear understanding of the process and the underlying principles.

Understanding the Distributive Property

At its core, the distributive property states that multiplying a number by a sum or difference inside parentheses is equivalent to multiplying the number by each term within the parentheses individually and then performing the addition or subtraction. Mathematically, this can be represented as:

  • a(b + c) = ab + ac
  • a(b - c) = ab - ac

Here, 'a' is the number being multiplied, and 'b' and 'c' are the terms inside the parentheses. The distributive property allows us to break down the expression, making it easier to manage and simplify.

To truly grasp the concept, let's consider a practical example. Imagine you're buying 3 boxes of pencils, and each box contains 10 regular pencils and 5 colored pencils. To find the total number of pencils, you could either add the number of pencils in one box (10 + 5 = 15) and then multiply by 3 (3 * 15 = 45), or you could multiply the number of regular pencils by 3 (3 * 10 = 30) and the number of colored pencils by 3 (3 * 5 = 15) and then add the results (30 + 15 = 45). Both methods yield the same answer, illustrating the essence of the distributive property.

Applying the Distributive Property to 9(8x - 5)

Now, let's apply the distributive property to the expression 9(8x - 5). Here, 9 is the number being multiplied, and (8x - 5) is the expression inside the parentheses. Following the property, we need to multiply 9 by each term within the parentheses individually:

  1. Multiply 9 by 8x: 9 * 8x = 72x
  2. Multiply 9 by -5: 9 * -5 = -45

Next, we combine these results to form the simplified expression:

72x - 45

Therefore, by applying the distributive property, we have successfully simplified the expression 9(8x - 5) to 72x - 45. This simplified form is much easier to work with in further calculations or algebraic manipulations.

Step-by-Step Breakdown

To ensure clarity, let's break down the process into a step-by-step format:

Step 1: Identify the terms. In the expression 9(8x - 5), identify the number outside the parentheses (9) and the terms inside the parentheses (8x and -5).

Step 2: Distribute the multiplication. Multiply the number outside the parentheses (9) by each term inside the parentheses:

  • 9 * 8x
  • 9 * -5

Step 3: Perform the multiplication. Calculate the products:

  • 9 * 8x = 72x
  • 9 * -5 = -45

Step 4: Combine the results. Write the results as a simplified expression:

72x - 45

By following these steps, you can confidently apply the distributive property to simplify various expressions.

Common Mistakes to Avoid

While the distributive property is relatively straightforward, there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid them:

  1. Forgetting to distribute to all terms: A common error is to multiply the number outside the parentheses by only the first term inside, neglecting the other terms. Remember, the distributive property requires you to multiply by every term within the parentheses.

  2. Incorrectly handling negative signs: Pay close attention to negative signs when distributing. For example, in the expression 9(8x - 5), you need to multiply 9 by both 8x and -5. A missed negative sign can lead to an incorrect result.

  3. Combining unlike terms: After applying the distributive property, you may end up with terms that cannot be combined. For instance, in the simplified expression 72x - 45, 72x and -45 are unlike terms (one has a variable, and the other is a constant) and cannot be added or subtracted.

By carefully avoiding these common mistakes, you can ensure accurate application of the distributive property.

Practice Problems

To solidify your understanding of the distributive property, let's work through a few practice problems:

  1. Simplify: 4(3y + 2)
  2. Simplify: -2(5z - 1)
  3. Simplify: 6(2a + 3b)

Solutions:

  1. 4(3y + 2) = 4 * 3y + 4 * 2 = 12y + 8
  2. -2(5z - 1) = -2 * 5z + (-2) * (-1) = -10z + 2
  3. 6(2a + 3b) = 6 * 2a + 6 * 3b = 12a + 18b

By practicing these problems, you'll gain confidence in your ability to apply the distributive property effectively.

Real-World Applications

The distributive property isn't just a mathematical concept confined to textbooks. It has numerous real-world applications. Let's explore a couple of examples:

  1. Calculating the cost of multiple items: Imagine you're buying 5 notebooks, and each notebook costs $2.50 plus a tax of $0.25. You can use the distributive property to calculate the total cost: 5($2.50 + $0.25) = 5 * $2.50 + 5 * $0.25 = $12.50 + $1.25 = $13.75

  2. Determining the area of a composite shape: Suppose you have a rectangular garden with a length of 10 feet and a width of (x + 3) feet. The area of the garden can be calculated using the distributive property: 10(x + 3) = 10 * x + 10 * 3 = 10x + 30 square feet.

These examples demonstrate how the distributive property can be applied in practical situations to solve everyday problems.

Conclusion

In conclusion, the distributive property is a powerful tool for simplifying expressions involving parentheses and multiplication. By understanding its principles and practicing its application, you can confidently tackle a wide range of mathematical problems. Remember to distribute the multiplication to every term within the parentheses, pay attention to negative signs, and avoid combining unlike terms. With consistent practice, you'll master the distributive property and enhance your overall mathematical proficiency.

This guide has provided a comprehensive explanation of the distributive property, its application to the expression 9(8x - 5), common mistakes to avoid, practice problems, and real-world applications. By mastering this fundamental concept, you'll be well-equipped to tackle more complex algebraic challenges.