Factoring Common Factors: Step-by-Step Calculations

by SLV Team 52 views

Hey guys! Let's dive into the world of factoring common factors. This is a super useful technique in math that makes calculations much easier. We're going to break down six different expressions step by step, so you can see exactly how it works. Get ready to boost your math skills!

Understanding Factoring Common Factors

So, what's the deal with factoring common factors? Basically, it's like finding a shared ingredient in a recipe. When you spot a number that multiplies into multiple terms, you can pull it out and simplify the expression. This not only makes the math less scary but also gives you a clearer picture of what's going on. For instance, in the expression 8 * 15 + 8 * 25, the number 8 is the common factor. By extracting it, we can make the calculation much simpler. Factoring is super handy because it reduces the complexity of calculations and makes them more manageable. It’s a foundational skill that you’ll use again and again in algebra and beyond, helping you to solve equations, simplify expressions, and even tackle real-world problems more efficiently. Think of it as your mathematical superpower for simplifying things!

Factoring common factors is also essential for solving more complex algebraic problems. When you encounter equations with multiple terms, identifying and factoring out common elements can significantly reduce the equation's complexity. This simplification process often reveals the underlying structure of the equation, making it easier to find solutions. Moreover, it's not just about simplifying numbers; factoring helps in understanding the relationships between variables and constants. In essence, mastering this skill is like equipping yourself with a mathematical Swiss Army knife – it's versatile, practical, and incredibly useful in a variety of situations. So, let's get started and see how this works in practice with some real examples!

Factoring common factors isn't just a mathematical trick; it's a way of thinking that promotes clarity and efficiency. When you factor, you're essentially reorganizing an expression to highlight its fundamental components. This reorganization can be particularly useful in fields like computer science and engineering, where optimizing processes and resource allocation is critical. In computer programming, for example, factoring can help streamline code, making it more efficient and easier to maintain. Similarly, in engineering, it can simplify complex calculations, allowing for more accurate and faster problem-solving. The ability to identify and extract common factors is, therefore, a skill that extends far beyond the classroom, equipping you with a powerful tool for tackling a wide range of challenges in various domains.

Let's Calculate!

Now, let's get into the calculations themselves. We’ll go through each expression step-by-step, so you can follow along easily.

a) 8 * 15 + 8 * 25

First up, we have 8 * 15 + 8 * 25. See that 8 appearing in both parts? That's our common factor!

  • Step 1: Extract the common factor (8). We rewrite the expression as 8 * (15 + 25). We've essentially pulled the 8 out, and now it's multiplying the sum of 15 and 25.
  • Step 2: Calculate the sum inside the parentheses. 15 + 25 equals 40. So, our expression now looks like 8 * 40.
  • Step 3: Multiply. 8 times 40 is 320.

So, 8 * 15 + 8 * 25 = 320. See how much simpler that was than doing each multiplication separately and then adding?

b) 5 * 54 - 5 * 24

Next, we have 5 * 54 - 5 * 24. This time, our common factor is 5.

  • Step 1: Extract the common factor (5). We rewrite the expression as 5 * (54 - 24). We've factored out the 5, which now multiplies the difference between 54 and 24.
  • Step 2: Calculate the difference inside the parentheses. 54 - 24 equals 30. So, our expression becomes 5 * 30.
  • Step 3: Multiply. 5 times 30 is 150.

Thus, 5 * 54 - 5 * 24 = 150. Factoring makes subtraction a breeze too!

c) 4 * 52 + 4 * 18

Moving on to 4 * 52 + 4 * 18, the common factor here is 4.

  • Step 1: Extract the common factor (4). We rewrite the expression as 4 * (52 + 18). We’ve pulled out the 4, and it’s now multiplying the sum of 52 and 18.
  • Step 2: Calculate the sum inside the parentheses. 52 + 18 equals 70. So, our expression now looks like 4 * 70.
  • Step 3: Multiply. 4 times 70 is 280.

Therefore, 4 * 52 + 4 * 18 = 280. Notice how extracting the common factor simplifies the calculation process significantly. By breaking down the problem into smaller, more manageable steps, we avoid the need for cumbersome multiplication and addition, making the entire calculation more efficient and less prone to errors. This method is particularly useful when dealing with larger numbers, as it reduces the risk of making mistakes during manual calculations. Moreover, understanding and applying the common factor extraction method not only improves arithmetic skills but also lays a solid foundation for more advanced algebraic concepts. It’s a skill that pays dividends in various mathematical contexts, making it an essential tool in any math student’s arsenal. So, keep practicing and you'll become a factoring pro in no time!

d) 7 * 46 + 7 * 34

For 7 * 46 + 7 * 34, the common factor is 7. Let's break it down:

  • Step 1: Extract the common factor (7). We rewrite the expression as 7 * (46 + 34). We've pulled the 7 out, and it's now multiplying the sum of 46 and 34.
  • Step 2: Calculate the sum inside the parentheses. 46 + 34 equals 80. So, our expression now looks like 7 * 80.
  • Step 3: Multiply. 7 times 80 is 560.

So, 7 * 46 + 7 * 34 = 560. Factoring really shines when dealing with larger numbers, doesn't it?

e) 8 * 62 - 8 * 42

Now, let's tackle 8 * 62 - 8 * 42. Our common factor is 8.

  • Step 1: Extract the common factor (8). We rewrite the expression as 8 * (62 - 42). We've factored out the 8, which now multiplies the difference between 62 and 42.
  • Step 2: Calculate the difference inside the parentheses. 62 - 42 equals 20. So, our expression becomes 8 * 20.
  • Step 3: Multiply. 8 times 20 is 160.

Thus, 8 * 62 - 8 * 42 = 160. Notice how consistently this method simplifies the calculations. The beauty of factoring lies in its ability to transform complex expressions into manageable components, making arithmetic operations much less daunting. This approach not only reduces the mental load required for calculations but also minimizes the potential for errors, especially when dealing with numbers that are not easily handled mentally. By breaking down the problem into smaller, sequential steps, we gain a clearer understanding of the mathematical relationships at play, which ultimately enhances our problem-solving skills. So, keep practicing this method, and you'll find that it becomes second nature, making math a lot more accessible and enjoyable!

f) 6 * 75 - 6 * 25

Last but not least, we have 6 * 75 - 6 * 25. The common factor here is 6.

  • Step 1: Extract the common factor (6). We rewrite the expression as 6 * (75 - 25). We’ve pulled out the 6, and it’s now multiplying the difference between 75 and 25.
  • Step 2: Calculate the difference inside the parentheses. 75 - 25 equals 50. So, our expression now looks like 6 * 50.
  • Step 3: Multiply. 6 times 50 is 300.

Therefore, 6 * 75 - 6 * 25 = 300. What a neat way to simplify things, right?

Wrapping Up

So, there you have it! We've calculated six different expressions by factoring out the common factor. Remember, this technique is all about making math easier and more manageable. By identifying and extracting the common factor, you can simplify complex expressions into smaller, more digestible parts. This not only reduces the risk of errors but also enhances your understanding of the underlying mathematical principles. Factoring is a foundational skill that extends far beyond these examples, so keep practicing and you'll find it becomes second nature. With this tool in your math toolkit, you'll be well-equipped to tackle a wide range of problems with confidence and ease. Keep up the great work, guys, and happy calculating!