Solving 60÷{2×[-7+18÷(-3+12)]}-[7×(-3)-18÷(-2)+1] A Step-by-Step Guide
Hey guys! Ever stumbled upon a math problem that looks like it belongs more in a spaceship control panel than on a piece of paper? Well, you're not alone! Today, we're going to break down a beast of a problem: 60÷{2×[-7+18÷(-3+12)]}-[7×(-3)-18÷(-2)+1]. This looks intimidating, but trust me, with a little step-by-step guidance, we can conquer it together. Think of it like a puzzle – each piece (or operation) has its place, and once we put them together in the right order, the solution will magically appear. So, grab your pencils, put on your thinking caps, and let’s dive into the world of mathematical gymnastics! We'll be using the order of operations, often remembered by the acronym PEMDAS (or BODMAS), which stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This is our roadmap to success, ensuring we tackle each operation in the correct sequence, like a well-choreographed dance. Before we even start crunching numbers, let's take a moment to appreciate the beauty of structured problem-solving. Math isn't just about getting the right answer; it's about developing a logical and methodical approach. These skills translate far beyond the classroom, helping us in everyday decision-making and problem-solving scenarios. Now, let's transform this mathematical monster into a series of manageable steps, making it less daunting and more approachable. Remember, there’s no rush! Take your time, double-check your work, and enjoy the process of unraveling the solution. Each step we complete is a victory, and by the end, we'll not only have the answer but also a deeper understanding of how mathematical operations work together. Let’s get started and turn this complex equation into a walk in the park!
Understanding the Order of Operations (PEMDAS/BODMAS)
Before we even think about touching the numbers in our equation, let’s quickly revisit our trusty friend, PEMDAS (or BODMAS if you learned it that way). This acronym is the golden rule of mathematical operations, telling us the exact order in which we need to solve the problem. PEMDAS stands for Parentheses (or Brackets), Exponents (or Orders), Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Think of it as a hierarchical structure where each level needs to be cleared before moving on to the next. Parentheses/Brackets are our first priority. They're like VIP sections in a club – everything inside them gets resolved before anything outside. This is because they often group operations that need to be treated as a single unit. Inside the parentheses, we again follow the PEMDAS rule, ensuring that we maintain the correct order within the group. Next up are Exponents (or Orders), which involve powers and roots. These guys indicate repeated multiplication and are handled after parentheses. Think of squaring a number – it’s a more powerful operation than simple multiplication or addition. Then comes the dynamic duo: Multiplication and Division. These two operations hold equal weight and are performed from left to right. This is crucial because changing the order can drastically alter the result. Imagine dividing before multiplying when you shouldn't – it's like changing lanes without checking your mirrors! Similarly, Addition and Subtraction are the final steps, also performed from left to right. They are the foundational operations, bringing everything together to give us the final answer. The reason we need this order is to ensure consistency and clarity in mathematical calculations. Without a standardized system, the same equation could yield different results depending on who's solving it. PEMDAS acts as our universal language, allowing mathematicians (and anyone else using numbers) to communicate effectively and accurately. By following this order, we maintain the integrity of the math and arrive at the one true answer. So, remember PEMDAS – it's not just a catchy acronym; it's the key to unlocking the secrets of complex equations. With this rule firmly in our minds, let’s return to our original problem and begin dissecting it, step by meticulous step.
Step 1: Simplify Inside the Innermost Parentheses (-3+12)
Okay, with PEMDAS as our guiding star, we’re ready to tackle our complex equation. The first order of business, as PEMDAS dictates, is to conquer the parentheses. But wait! We have parentheses nested within parentheses, like a set of Russian dolls. So, where do we start? Always, always start with the innermost set. It's like peeling an onion – you start with the outer layers to get to the core. Looking at our equation, 60÷2×[-7+18÷(-3+12)]}-[7×(-3)-18÷(-2)+1]**, the innermost parentheses contain the simple operation (-3 + 12). This is a friendly addition problem, a welcome sight amidst the larger equation. To solve this, we simply add -3 and 12. Think of it as starting at -3 on a number line and moving 12 steps to the right. Where do we end up? At 9! So, (-3 + 12) = 9. That wasn't so bad, was it? We've successfully navigated the first hurdle. Now, let's rewrite our equation with this simplification. Our equation now looks like this-[7×(-3)-18÷(-2)+1]. See how much cleaner it already looks? By focusing on one small step at a time, we’re making this complex problem much more manageable. This principle of breaking down a large task into smaller, digestible steps is a valuable skill not just in math but in life in general. Whether it’s planning a project, learning a new skill, or even just cleaning your room, breaking it down makes the overwhelming seem achievable. Now that we've simplified the innermost parentheses, we can move on to the next set, still following the PEMDAS guide. But before we jump ahead, let's take a moment to appreciate this small victory. We've identified the starting point, executed the operation, and updated our equation. This methodical approach is what will lead us to the final solution. Onward and upward! The parentheses are calling, and we have more simplifying to do. Let’s continue to chip away at this equation, one step at a time, until we reach the satisfying conclusion.
Step 2: Perform Division Within the Brackets (18÷9)
Alright, we’ve conquered the innermost parentheses, and now it’s time to venture further into the mathematical jungle. Our equation currently stands as: 60÷2×[-7+18÷9]}-[7×(-3)-18÷(-2)+1]**. Following PEMDAS, we need to stay within the brackets and address any remaining operations inside them. We spot a division operation within the first set of brackets-[7×(-3)-18÷(-2)+1]. Notice how each step of simplification makes the equation less intimidating and more approachable. We started with a complex jumble of numbers and operations, and now it’s gradually morphing into something much clearer. This is the power of methodical problem-solving – breaking down the complex into manageable chunks. Before we move on, let's take a moment to appreciate how far we've come. We identified the correct operation, performed it accurately, and updated our equation. This process of identify, execute, and update is crucial for tackling any mathematical problem, especially the more challenging ones. Now, with the division within the brackets resolved, we can move on to the next operation within those same brackets. Remember, we're still working our way from the inside out, peeling those layers like an onion. The brackets still hold some secrets, and it's our mission to uncover them. Let's keep our PEMDAS compass pointing in the right direction and continue our journey through this equation. The end is in sight, and we're making excellent progress!
Step 3: Solve the Addition/Subtraction Inside the Brackets [-7+2]
We're on a roll, guys! Our equation is steadily shrinking as we methodically tackle each operation. Currently, we have: 60÷2×[-7+2]}-[7×(-3)-18÷(-2)+1]**. Sticking with PEMDAS, we're still focusing on the brackets. Inside the first set of brackets, we now have an addition operation-[7×(-3)-18÷(-2)+1]. We’ve simplified the expression within the first set of brackets down to a single number, -5. This is a significant milestone! It means we've successfully navigated all the additions and subtractions within those brackets. Remember, the key to solving complex problems is to take them one step at a time. We didn't try to jump ahead or skip any steps; we followed the order of operations meticulously. This approach not only ensures accuracy but also helps build a strong understanding of the underlying mathematical principles. Before we move on, let’s take a moment to acknowledge our progress. We started with a complex expression within brackets and have now reduced it to a single, manageable number. This demonstrates the power of PEMDAS and the effectiveness of our step-by-step approach. Now, we're ready to move on to the next operation, which involves multiplication within the curly braces. The finish line is getting closer, and we're maintaining a steady pace. Let’s keep our focus, continue to follow the PEMDAS roadmap, and watch this equation transform into its final solution.
Step 4: Multiply Inside the Curly Braces {2×[-5]}
Great job, team! We're making excellent progress on our mathematical adventure. Our equation currently stands as: 60÷2×[-5]}-[7×(-3)-18÷(-2)+1]**. Following our trusty PEMDAS guide, we need to tackle the operations within the curly braces. Inside these braces, we have a multiplication operation-[7×(-3)-18÷(-2)+1]. See how the curly braces now contain a single number? We've effectively simplified that entire expression! This is a testament to our methodical approach and our commitment to following the order of operations. Before we proceed, let’s pause for a moment to appreciate the elegance of this process. We started with a complex expression within the curly braces and, through careful application of PEMDAS, have reduced it to a simple number. This highlights the power of structured problem-solving and the importance of breaking down large tasks into smaller, manageable steps. Now, we're ready to move on to the next set of operations, which reside within the square brackets. The square brackets contain a series of multiplications, divisions, additions, and subtractions, so we’ll need to carefully apply PEMDAS within that section as well. The journey isn't over yet, but we're definitely on the right track. Let's keep our momentum going and continue to unravel this equation, one operation at a time. The solution is within reach, and we're determined to find it!
Step 5: Simplify Inside the Square Brackets [7×(-3)-18÷(-2)+1]
Fantastic work, everyone! We're truly becoming mathematical masters. Our equation is now: 60÷-10}-[7×(-3)-18÷(-2)+1]**. Our next mission, guided by PEMDAS, is to simplify the expression within the square brackets-[-11]. We've made significant progress in simplifying the original complex equation. By methodically applying PEMDAS and breaking down the problem into smaller, manageable steps, we’ve navigated through the operations within the square brackets. Before we move on, let’s take a moment to celebrate this achievement. We tackled a series of operations within the brackets, carefully following the order of operations, and arrived at a single number. This demonstrates our growing mathematical prowess and our ability to handle complex calculations. Now, we're ready to address the remaining operations in the equation. The finish line is within sight, and we're determined to cross it. Let’s keep our focus, continue to apply PEMDAS, and watch as the final solution emerges.
Step 6: Perform the Division 60÷{-10}
Excellent work, team! We're in the home stretch now. Our equation has been simplified to: 60÷-10}-[-11]**. According to PEMDAS, we need to address division before subtraction. So, let's tackle the division operation. Dividing a positive number by a negative number gives us a negative result. So, 60 divided by -10 is -6. Thus, 60 ÷ {-10} = -6. We’ve successfully executed another operation and are getting closer to the final answer. Let's update our equation to reflect this simplification. The equation now becomes: -6 - [-11]. Notice how our equation is becoming increasingly streamlined. We've systematically eliminated the parentheses, brackets, and braces, and are now left with a simple subtraction problem. This is a testament to the power of methodical problem-solving and our unwavering commitment to PEMDAS. Before we proceed, let’s take a moment to acknowledge how far we’ve come. We started with a complex jumble of numbers and operations, and through careful application of the order of operations, we’ve transformed it into a straightforward equation. This demonstrates our growing mathematical confidence and our ability to tackle challenging problems. Now, we're ready for the final step: the subtraction. We're on the verge of discovering the solution, and the excitement is building. Let’s keep our focus, apply the last remaining operation, and claim our victory!
Step 7: Final Subtraction -6 - [-11]
We've reached the final step, everyone! Our equation is beautifully simplified to: -6 - [-11]. This is a subtraction operation, but it involves subtracting a negative number. Remember, subtracting a negative number is the same as adding its positive counterpart. So, -6 - [-11] is the same as -6 + 11. Now, we have a simple addition problem. Think of it as starting at -6 on a number line and moving 11 steps to the right. Where do we end up? At 5! So, -6 + 11 = 5. Therefore, -6 - [-11] = 5. We’ve done it! We’ve successfully navigated through the complex equation and arrived at the final solution. The answer is 5. Let’s take a moment to celebrate this achievement! We started with a seemingly daunting problem, filled with parentheses, brackets, braces, and a variety of operations. But, by systematically applying the order of operations (PEMDAS) and breaking down the problem into smaller, manageable steps, we conquered it. This journey has not only provided us with the answer but has also strengthened our problem-solving skills and our understanding of mathematical principles. Before we conclude, let’s reflect on the process. We began by understanding the importance of PEMDAS, then we meticulously worked through each step, simplifying the equation along the way. We encountered various operations, from addition and subtraction to multiplication and division, and we handled them all with precision and confidence. This is the power of a structured approach and the beauty of mathematics. Congratulations, everyone! We’ve successfully solved the equation 60÷{2×[-7+18÷(-3+12)]}-[7×(-3)-18÷(-2)+1], and the answer is 5. Go forth and conquer more mathematical challenges!
Conclusion
So, guys, we did it! We successfully solved the complex equation 60÷{2×[-7+18÷(-3+12)]}-[7×(-3)-18÷(-2)+1], and the answer is 5. What a journey! We started with a problem that looked like a mathematical monster, but by breaking it down step-by-step and diligently following the order of operations (PEMDAS), we tamed the beast and emerged victorious. This exercise wasn't just about finding the right answer; it was about reinforcing the importance of methodical problem-solving. We learned that even the most intimidating challenges can be overcome by breaking them into smaller, manageable steps. Think about it – this approach applies to so much more than just math. Whether you're tackling a big project at work, learning a new skill, or even just organizing your closet, the principle of breaking it down makes the overwhelming feel achievable. We also honed our understanding of PEMDAS, that trusty acronym that guides us through the maze of mathematical operations. Parentheses, Exponents, Multiplication and Division, Addition and Subtraction – these aren't just random words; they're the roadmap to mathematical success. By mastering the order of operations, we can ensure accuracy and consistency in our calculations. But perhaps the most important takeaway from this exercise is the confidence we've gained. We faced a tough problem head-on, we persevered through each step, and we ultimately found the solution. This experience demonstrates our ability to learn, adapt, and overcome challenges – skills that will serve us well in all aspects of life. So, the next time you encounter a complex problem, remember our journey today. Remember the power of PEMDAS, the importance of breaking things down, and the confidence you gained from conquering this mathematical beast. And most importantly, remember that math can be fun! It's a puzzle to be solved, a code to be cracked, and a challenge to be embraced. Keep practicing, keep exploring, and keep pushing your mathematical boundaries. The world of numbers is vast and fascinating, and there's always more to discover. Until next time, happy solving!
