Solve: 6 − [4 − 3(4 − 2)] − {7 − 5 [4 − 2(7 − 1)]}

by Rajiv Sharma 51 views

Let's dive into breaking down this complex mathematical expression step by step. I know it might look intimidating at first glance, but don't worry, we'll tackle it together! We're going to use the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), to solve this. Think of PEMDAS as our roadmap; it tells us exactly which operations to perform in what order. So, grab your pencils, and let's get started!

Understanding the Order of Operations (PEMDAS)

Before we even think about touching the actual numbers, it's crucial to understand the golden rule of math: the order of operations, also known as PEMDAS. PEMDAS is our guiding principle ensuring we solve expressions consistently and accurately. It's the universal language of math, preventing chaos and confusion. So, what does PEMDAS actually stand for? Let's break it down:

  • Parentheses: This is where we start. Any operation inside parentheses (or brackets) gets our immediate attention. It's like a VIP section in the math world – what's inside gets priority.
  • Exponents: Next up are exponents, those little superscript numbers that tell us to multiply a number by itself a certain number of times. Think of them as shorthand for repeated multiplication. For instance, 2³ (2 to the power of 3) means 2 * 2 * 2.
  • Multiplication and Division: These two are like partners in crime; they have equal priority. We work them from left to right, just like reading a sentence. If division comes before multiplication in the expression (reading from left to right), we do division first, and vice versa. Remember this left-to-right rule!
  • Addition and Subtraction: Last but not least, we have addition and subtraction. Just like multiplication and division, they have equal priority, and we solve them from left to right. So, if subtraction appears before addition, we subtract first.

Why is PEMDAS so important, you ask? Well, imagine if we all just randomly performed operations in any order we pleased. We'd get a mishmash of different answers, and math would be a chaotic mess! PEMDAS brings structure and consistency, ensuring everyone arrives at the same solution for the same problem. It’s the foundation upon which more advanced mathematical concepts are built. Ignoring PEMDAS is like trying to build a house without a blueprint – it might look okay at first, but it's bound to crumble eventually. So, let's always keep PEMDAS in mind as we tackle mathematical expressions; it's our best friend in the math world.

Breaking Down the Expression Step-by-Step

Now, let's get our hands dirty and solve the expression: 6 − [4 − 3(4 − 2)] − {7 − 5 [4 − 2(7 − 1)]}. Guys, don't let the brackets and braces scare you! We'll work from the inside out, following our trusty PEMDAS guide.

  1. Innermost Parentheses: Start with the innermost parentheses: (4 − 2) and (7 − 1).

    • (4 − 2) = 2
    • (7 − 1) = 6

    Our expression now looks like this: 6 − [4 − 3(2)] − {7 − 5 [4 − 2(6)]}.

  2. Next Level Parentheses: Now we focus on the expressions within the square brackets [ ] and curly braces { }.

    • Inside the square brackets: 3(2) multiplication comes first.

      • 3(2) = 6

      • So, [4 − 3(2)] becomes [4 − 6]

      • [4 − 6] = -2

    • Inside the curly braces, we have [4 − 2(6)] first.

      • 2(6) = 12

      • So, [4 − 2(6)] becomes [4 − 12]

      • [4 − 12] = -8

    • Now, let's look at the entire expression inside the curly braces: {7 − 5 [4 − 2(6)]}, which simplifies to {7 − 5(-8)}

    • Multiply: 5(-8) = -40

    • So, {7 − 5(-8)} becomes {7 − (-40)}

    • {7 − (-40)} = {7 + 40} = 47

    Now our expression looks much simpler: 6 − (-2) − 47.

  3. Final Operations: Finally, we're left with subtractions. Remember to work from left to right.

    • 6 − (-2) is the same as 6 + 2 = 8

    • So, the expression becomes 8 − 47

    • 8 − 47 = -39

    Therefore, the solution to the complex expression 6 − [4 − 3(4 − 2)] − {7 − 5 [4 − 2(7 − 1)]} is -39. Isn't it satisfying to see it all come together?

Common Mistakes to Avoid

Solving complex mathematical expressions can be tricky, and it's easy to make mistakes if you're not careful. But don't worry, guys, I'm here to help you dodge those common pitfalls! Let's take a look at some frequent errors and how to steer clear of them. Knowing what mistakes to avoid is half the battle.

  1. Ignoring the Order of Operations (PEMDAS): This is the number one culprit behind math errors. It's so crucial that we talked about it extensively earlier. It’s important, so let's emphasize it again! Jumping the gun and performing operations in the wrong order can throw everything off. For instance, doing addition before multiplication, or subtraction before dealing with parentheses. Remember, PEMDAS is our guiding star; it tells us exactly what to do and when. If you're ever unsure, just write out PEMDAS at the top of your paper as a little reminder. Think of it as your mathematical GPS, keeping you on the right track.

  2. Forgetting the Negative Sign: Negative signs can be sneaky little devils. It's super easy to lose track of them, especially when dealing with multiple operations and parentheses. A misplaced or forgotten negative sign can completely change the outcome of your calculation. Pay extra attention when you're subtracting negative numbers, or when you're multiplying or dividing by a negative number. A helpful trick is to rewrite subtraction as addition of a negative number. For example, instead of 5 − (-3), write 5 + 3. This can help you visualize the operation more clearly and reduce the risk of sign errors.

  3. Distributing Incorrectly: When you have a number multiplying a set of parentheses (like 3(x + 2)), you need to distribute that number to everything inside the parentheses. That means multiplying the 3 by both the x and the 2. A common mistake is to only multiply by the first term and forget the others. Remember, distribution is like sharing – everyone inside the parentheses gets a piece! Draw those little arrows to remind yourself to distribute properly; they're like visual cues that guide your hand and mind.

  4. Simplifying Too Quickly: We all want to get to the answer fast, but sometimes rushing can lead to silly mistakes. Take your time and simplify step-by-step, especially when the expression is complex. Write out each step clearly, so you can easily check your work and spot any errors. It might seem tedious at first, but it's way better than having to redo the whole problem because of a careless mistake.

  5. Not Checking Your Work: This is the final, and perhaps the most crucial, step. Once you've arrived at an answer, don't just assume it's correct. Take a few minutes to go back through your work and double-check each step. Did you follow PEMDAS? Did you distribute correctly? Did you keep track of your negative signs? It's like proofreading a paper before you submit it – a quick check can catch those little errors that you might have missed the first time around. There is also a good opportunity to use the calculator to confirm each operation.

By being aware of these common pitfalls and taking the time to avoid them, you'll be well on your way to conquering complex mathematical expressions like a pro! Remember, practice makes perfect, so keep at it, and don't be afraid to ask for help when you need it. Math can be challenging, but it's also incredibly rewarding when you crack a tough problem.

Conclusion

So, there you have it! We've successfully navigated the intricate world of this mathematical expression, demonstrating the power of PEMDAS and careful, step-by-step problem-solving. Remember, complex problems are just a series of smaller, manageable steps. Keep practicing, stay patient, and you'll be a math whiz in no time! Guys, I hope this breakdown has been helpful and has boosted your confidence in tackling similar challenges. Keep up the great work, and happy calculating!