Solving -[5-3x(2-10)+3÷(1-4)]: A Math Discussion

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Hey guys! Let's dive into this interesting mathematical expression: -[5-3x(2-10)+3÷(1-4)]. Math can sometimes look intimidating, but breaking it down step-by-step makes it super manageable. We're going to walk through each operation, making sure we follow the correct order (PEMDAS/BODMAS, remember?). Stick around, and by the end, you’ll be a pro at solving this kind of problem. Let's get started!

Understanding the Order of Operations

Before we even touch the numbers, it's crucial to understand the order of operations. Think of it as the golden rule of math! We use acronyms like PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) to remember the sequence. Basically, this order ensures we all arrive at the same answer, no matter who's solving the problem. So, what does this mean for our expression? It means we'll first tackle anything inside parentheses, then exponents (if any), followed by multiplication and division (from left to right), and finally, addition and subtraction (again, from left to right). This is super important, guys, so keep it in mind as we move forward!

Why is the order of operations so important? Imagine if we just went from left to right without any rules. We'd get a completely different answer! Following PEMDAS/BODMAS is like having a mathematical GPS; it guides us to the correct destination. Think about it: if we didn't agree on this order, math textbooks and calculators would be utter chaos. This standardized approach keeps everything consistent and reliable. Trust the process—it's been refined over years to make mathematical sense. We're not just blindly following rules; we're adhering to a system that maintains mathematical integrity and accuracy. So, next time you see a complex expression, remember PEMDAS/BODMAS—it's your best friend in the math world.

Step-by-Step Breakdown of the Expression

Okay, let's break down our expression -[5-3x(2-10)+3÷(1-4)] step by step. This is where the magic happens, and we start to see how everything fits together. First up: Parentheses/Brackets! We've got two sets to deal with: (2-10) and (1-4). Let's tackle them one at a time.

Step 1: Solving the Inner Parentheses

The first set, (2-10), is a straightforward subtraction. 2 minus 10 gives us -8. Easy peasy! Now, let's move to the second set, (1-4). Again, it's a simple subtraction. 1 minus 4 equals -3. So, our expression now looks like this: -[5-3x(-8)+3÷(-3)]. See how we've simplified the expression already? We're making progress, guys! Breaking down the initial parentheses is a foundational step, and it sets the stage for the rest of the calculation. Without tackling these first, we'd be swimming in a sea of numbers without a clear direction. So, always remember: Parentheses first! It's like setting up the base camp before climbing the mountain.

Step 2: Multiplication and Division

Next in line, according to PEMDAS/BODMAS, are Multiplication and Division. Remember, we perform these operations from left to right. In our updated expression, -[5-3x(-8)+3÷(-3)], we see both multiplication and division. First, we have -3 multiplied by -8. A negative times a negative equals a positive, so -3 x -8 = 24. Now our expression looks like -[5+24+3÷(-3)]. See how the expression is gradually simplifying? It’s like peeling back the layers of an onion, revealing the core bit by bit. This is why following the order of operations is so important—it helps us maintain clarity and accuracy.

Next up is the division: 3 divided by -3. A positive divided by a negative results in a negative, so 3 ÷ -3 = -1. Now our expression is looking even cleaner: -[5+24-1]. We've successfully handled the multiplication and division, and we're on the home stretch! Each step brings us closer to the final answer. Taking it one operation at a time allows us to manage the complexity and reduces the chances of making errors. We're doing great, guys! Just a little more to go.

Step 3: Addition and Subtraction

Now we're down to the final operations: Addition and Subtraction. Just like with multiplication and division, we work from left to right. Our expression currently reads -[5+24-1]. First, we add 5 and 24, which equals 29. So, the expression simplifies to -[29-1]. We’re almost there, guys! The numbers are getting smaller, and the solution is within reach. This stage is about consolidating all the previous steps into a final result. Each operation we've performed has built upon the last, guiding us to this point. It's like constructing a building, brick by brick, until we have a complete structure.

Now, we subtract 1 from 29, which gives us 28. So, inside the brackets, we have 28. Our expression now looks like -[28]. We've done all the heavy lifting, and we're just one step away from the final answer. This final step highlights the importance of precision throughout the entire process. If we had made a mistake earlier, it would have carried through to this point, affecting our result. But by diligently following PEMDAS/BODMAS, we've navigated the expression successfully.

Step 4: Applying the Negative Sign

Finally, we have -[28]. This simply means we take the negative of 28, which is -28. And that's it! We've solved the expression. Give yourselves a pat on the back, guys! We started with a seemingly complex problem and, by breaking it down step by step, arrived at the solution. This is the power of understanding and applying the order of operations. It’s not just about getting the right answer; it’s about developing a systematic approach to problem-solving. This skill will be invaluable, not just in math but in many areas of life.

The Final Answer and Key Takeaways

So, the final answer to the expression -[5-3x(2-10)+3÷(1-4)] is -28. Awesome job, everyone! We navigated through parentheses, multiplication, division, addition, and subtraction, and we nailed it. But more than just getting the answer, what did we learn along the way? Let's recap some key takeaways.

Key Takeaways

  1. PEMDAS/BODMAS is Your Best Friend: Always, always remember the order of operations. It's the foundation of solving any mathematical expression accurately. Whether it's a simple equation or a complex problem, PEMDAS/BODMAS will guide you. Think of it as your trusty map in the world of numbers. Without it, you might get lost, but with it, you're sure to reach your destination.
  2. Break It Down: Complex expressions can seem daunting at first, but breaking them down into smaller, manageable steps makes them much less intimidating. Just like we tackled each operation one at a time, you can apply this strategy to any challenging problem. Divide and conquer is the name of the game! Each step is a mini-victory, and they all add up to the final solution.
  3. Precision is Key: Every step counts. A small mistake early on can throw off the entire solution. So, take your time, double-check your work, and ensure accuracy at each stage. Math is like building a tower—a strong foundation is crucial. Each calculation must be precise to support the next, leading to a sturdy and correct final answer.
  4. Practice Makes Perfect: The more you practice, the more comfortable you'll become with these types of problems. Math is a skill, and like any skill, it improves with practice. Don't be afraid to tackle new and challenging expressions. Each problem you solve strengthens your understanding and builds your confidence.

Practice Problems

Want to put your skills to the test? Here are a couple of practice problems for you, guys:

  1. 2 + 3 x (6 - 4) ÷ 2
  2. -[10 - 2 x (5 + 1) + 4 ÷ (-2)]

Try solving these using the steps we discussed. Remember PEMDAS/BODMAS, break down the problem, and take your time. The answers are below, but try to solve them on your own first! Math is a journey, and each problem is a step forward. Embrace the challenge, and you'll be amazed at how much you can achieve.

Conclusion

So, there you have it, guys! We've successfully solved a complex mathematical expression and reinforced the importance of the order of operations. Remember, math isn't about memorizing formulas; it's about understanding the process and applying logic. Keep practicing, stay curious, and you'll become math whizzes in no time! Math is more than just numbers and equations; it's a way of thinking and a powerful tool for problem-solving in all areas of life. So, embrace the challenge and enjoy the journey!

Answers to Practice Problems:

  1. 2 + 3 x (6 - 4) ÷ 2 = 5
  2. -[10 - 2 x (5 + 1) + 4 ÷ (-2)] = 4