Simplifying [7(8-5)-6]-[3-(2-5)] A Step-by-Step Guide

Introduction to Order of Operations

In the realm of mathematics, mastering the order of operations is akin to learning the grammar of a language. Just as grammar dictates the structure of sentences, the order of operations ensures that mathematical expressions are evaluated consistently and unambiguously. Without a clear set of rules, the same expression could yield multiple results, leading to confusion and errors. The universally accepted convention, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), provides a roadmap for simplifying complex mathematical statements. This article delves into the intricacies of the order of operations, using the expression [7(8-5)-6]-[3-(2-5)] as a practical example to illustrate each step. By meticulously breaking down this expression, we'll not only arrive at the correct answer but also gain a deeper understanding of the underlying principles that govern mathematical calculations. Understanding and applying the order of operations correctly is not just a fundamental skill in mathematics; it is also crucial in various fields such as engineering, computer science, and economics, where complex calculations are commonplace. Therefore, a solid grasp of this concept is essential for anyone pursuing studies or careers in these areas. The expression [7(8-5)-6]-[3-(2-5)] may seem daunting at first glance, but by systematically applying the rules of PEMDAS, it can be simplified step by step. Each operation, from handling parentheses to performing subtraction, has its place in the sequence, ensuring that the final result is accurate and meaningful. This exploration will serve as a valuable exercise in mathematical precision and logical thinking. Let's embark on this journey of mathematical discovery, unraveling the layers of the expression [7(8-5)-6]-[3-(2-5)] and uncovering the elegance of the order of operations.

Step 1: Parentheses First

The first step in simplifying the expression [7(8-5)-6]-[3-(2-5)] is to address the parentheses. According to the order of operations, calculations within parentheses take precedence over all other operations. This is because parentheses group terms together, indicating that they should be treated as a single unit. In our expression, we encounter two sets of parentheses: (8-5) and (2-5). We'll tackle each one individually, starting with (8-5). This subtraction is straightforward: 8 minus 5 equals 3. So, we replace (8-5) with 3, and our expression now becomes [7(3)-6]-[3-(2-5)]. Next, we move to the second set of parentheses, (2-5). Here, we're subtracting a larger number from a smaller one, resulting in a negative number. 2 minus 5 equals -3. We replace (2-5) with -3, and the expression transforms into [7(3)-6]-[3-(-3)]. Notice how the parentheses are still present around the -3; this is crucial for maintaining clarity and ensuring that the subsequent operations are performed correctly. The parentheses around -3 serve as a reminder that this is a single, negative quantity. By simplifying the expressions within the parentheses first, we've taken a significant step towards unraveling the entire expression. This initial step not only reduces the complexity but also sets the stage for the remaining operations. Parentheses act as organizational tools in mathematical expressions, guiding us on which calculations to prioritize. As we continue to work through the expression, we'll see how each step builds upon the previous one, ultimately leading us to the final solution. The importance of correctly handling parentheses cannot be overstated; a mistake in this initial stage can propagate through the rest of the calculation, leading to an incorrect answer. Therefore, careful attention to detail is paramount. With the parentheses now simplified, we're ready to move on to the next level of operations, further simplifying the expression.

Step 2: Multiplication within Brackets

Having addressed the parentheses, the next step in simplifying the expression [7(3)-6]-[3-(-3)] is to tackle the multiplication within the brackets. According to the order of operations, multiplication takes precedence over addition and subtraction. In this case, we have 7(3) inside the first set of brackets. This notation signifies 7 multiplied by 3, which equals 21. Replacing 7(3) with 21, our expression now reads [21-6]-[3-(-3)]. It's important to note that we're focusing on the multiplication within the brackets first because the brackets act as a grouping symbol, similar to parentheses. Operations within brackets must be completed before operations outside of them. Now that we've performed the multiplication, the first set of brackets contains a simple subtraction: 21 minus 6. This simplification brings us closer to isolating a single numerical value for each set of brackets. The multiplication step not only reduces the complexity of the expression but also highlights the importance of adhering to the order of operations. If we had attempted to perform the subtraction before the multiplication, we would have arrived at an incorrect result. The consistent application of PEMDAS ensures that mathematical expressions are evaluated in a standardized manner, leading to accurate solutions. As we move forward, we'll continue to follow this established order, simplifying each component of the expression systematically. The brackets serve as visual cues, guiding us on which operations to prioritize. By focusing on the multiplication within the brackets, we've effectively reduced the number of operations required, making the expression more manageable. This step-by-step approach is crucial for solving complex mathematical problems, breaking them down into smaller, more digestible parts. With the multiplication complete, we're now poised to address the remaining operations within the brackets, further simplifying the expression and moving closer to the final answer.

Step 3: Subtraction within Brackets

Following the multiplication, we now focus on the subtraction within the brackets. Our expression currently stands as [21-6]-[3-(-3)]. Within the first set of brackets, we have 21-6, which is a straightforward subtraction. 21 minus 6 equals 15. So, we replace [21-6] with 15, and the expression becomes 15-[3-(-3)]. Now, let's turn our attention to the second set of brackets, [3-(-3)]. This requires careful consideration of the negative signs. Subtracting a negative number is equivalent to adding the positive counterpart. Therefore, 3-(-3) is the same as 3+3, which equals 6. Replacing [3-(-3)] with 6, our expression is now simplified to 15-6. By performing the subtractions within the brackets, we've eliminated the brackets altogether, reducing the expression to a simple subtraction problem. This step demonstrates the power of systematically applying the order of operations, breaking down a complex expression into manageable parts. The careful handling of negative signs is crucial in mathematics, and this step highlights the importance of understanding the rules governing their manipulation. Subtracting a negative number is a common source of errors, so it's essential to remember the principle that subtracting a negative is the same as adding a positive. With the brackets now completely resolved, we're left with a single operation: subtraction. This final step will lead us to the solution of the expression. The process of simplifying the expression step by step, from parentheses to subtraction, showcases the elegance and efficiency of the order of operations. Each operation has its place in the sequence, ensuring that the final result is accurate and meaningful. As we approach the final step, we can appreciate the progress we've made in unraveling the expression, transforming it from a seemingly complex problem into a simple calculation.

Step 4: Final Subtraction

With the brackets now fully resolved, we arrive at the final step in simplifying the expression. Our expression has been reduced to 15-6. This is a straightforward subtraction problem: 15 minus 6. Performing this subtraction, we find that 15 minus 6 equals 9. Therefore, the final simplified value of the expression [7(8-5)-6]-[3-(2-5)] is 9. This result represents the culmination of all the previous steps, each of which played a crucial role in arriving at the correct answer. The final subtraction serves as a testament to the power of the order of operations, demonstrating how a complex expression can be systematically broken down into simpler components. The journey from the initial expression to the final result highlights the importance of following the established rules of mathematics. Each operation, from parentheses to subtraction, was performed in the correct sequence, ensuring that the final answer is accurate and unambiguous. The value 9 is not just a number; it is the solution to a puzzle, a testament to our ability to apply mathematical principles and arrive at a logical conclusion. The process of simplifying the expression has not only yielded a numerical answer but has also reinforced our understanding of the order of operations and its significance in mathematical calculations. As we reflect on the steps we've taken, we can appreciate the elegance and efficiency of the mathematical framework that allows us to solve such problems. The final subtraction, the last piece of the puzzle, completes the picture, revealing the solution and affirming the validity of our approach. The result, 9, stands as a concise and definitive answer to the original expression, a testament to the power of mathematical reasoning and the consistent application of the order of operations.

Conclusion

In conclusion, the simplification of the expression [7(8-5)-6]-[3-(2-5)] to its final value of 9 exemplifies the critical role of the order of operations in mathematics. By meticulously following the PEMDAS convention, we navigated through parentheses, multiplication, and subtraction, step by step, to arrive at the correct answer. This exercise underscores the importance of adhering to established mathematical rules to ensure accuracy and consistency in calculations. The order of operations is not merely a set of guidelines; it is the foundation upon which complex mathematical expressions are built and solved. Without a clear understanding and application of these rules, the same expression could yield multiple different results, leading to confusion and errors. The expression [7(8-5)-6]-[3-(2-5)] served as a practical example to illustrate how PEMDAS works in action. We first addressed the parentheses, simplifying the expressions within them. Next, we tackled the multiplication, followed by the subtractions within the brackets. Finally, we performed the last subtraction, arriving at the solution of 9. Each step was crucial, and any deviation from the correct order would have led to an incorrect result. This detailed walkthrough not only demonstrates the mechanics of simplifying the expression but also reinforces the underlying principles of mathematical logic and precision. The ability to correctly apply the order of operations is a fundamental skill in mathematics, essential for success in algebra, calculus, and beyond. It is also a valuable skill in various real-world applications, from engineering and finance to computer science and data analysis. The expression [7(8-5)-6]-[3-(2-5)] may seem like a simple example, but it encapsulates the core concepts of mathematical operations and their proper sequence. By mastering these concepts, we equip ourselves with the tools to tackle more complex problems and navigate the world of mathematics with confidence. The final answer, 9, is not just a number; it is a testament to our understanding of mathematical principles and our ability to apply them effectively.