Solving (5-3)x[(-6)+(5+1)-(-5)]-[(5+6)-(-7+9)-(-2)] A Step-by-Step Guide
In the realm of mathematics, numerical expressions often appear daunting at first glance. However, by methodically applying the order of operations and simplifying each component, even the most complex expressions can be deciphered. This article delves into the step-by-step solution of the numerical expression (5-3)x[(-6)+(5+1)-(-5)]-[(5+6)-(-7+9)-(-2)], providing a comprehensive guide for students and enthusiasts alike. This seemingly intricate expression, (5-3)x[(-6)+(5+1)-(-5)]-[(5+6)-(-7+9)-(-2)], can be systematically solved by adhering to the fundamental principles of mathematical operations. By breaking down the expression into manageable parts and following the established order of operations, we can arrive at the solution with clarity and precision. This guide aims to illuminate the process, making it accessible and understandable for learners of all levels. The ability to solve numerical expressions like (5-3)x[(-6)+(5+1)-(-5)]-[(5+6)-(-7+9)-(-2)] is not merely an academic exercise; it forms the cornerstone of various mathematical disciplines and real-world applications. From calculating financial transactions to designing engineering structures, the understanding of order of operations and simplification is paramount. This article serves as a valuable resource for honing these essential skills and building a solid foundation in mathematics. The journey of solving (5-3)x[(-6)+(5+1)-(-5)]-[(5+6)-(-7+9)-(-2)] is not just about arriving at the correct answer; it's about developing a logical and systematic approach to problem-solving. This approach, once mastered, can be applied to a wide range of challenges, both within and beyond the realm of mathematics. So, let's embark on this mathematical adventure and unlock the secrets hidden within this expression. The step-by-step breakdown will not only demystify the process but also empower you with the confidence to tackle similar problems with ease.
Decoding the Expression: A Step-by-Step Approach
To effectively solve the numerical expression (5-3)x[(-6)+(5+1)-(-5)]-[(5+6)-(-7+9)-(-2)], we must adhere to the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This rule dictates the sequence in which we perform the operations to ensure a consistent and accurate result. Let's begin by tackling the innermost parentheses within the expression (5-3)x[(-6)+(5+1)-(-5)]-[(5+6)-(-7+9)-(-2)]. This initial step is crucial as it sets the stage for the subsequent simplifications. By focusing on the innermost elements first, we break down the complexity and make the expression more manageable. The order of operations is not just a rule; it's a roadmap for navigating the intricate world of mathematical expressions. Without it, the same expression could yield multiple different answers, leading to confusion and errors. Therefore, understanding and applying PEMDAS is fundamental to achieving mathematical proficiency. As we progress through the steps, we'll see how each operation builds upon the previous one, ultimately leading us to the final solution. The expression (5-3)x[(-6)+(5+1)-(-5)]-[(5+6)-(-7+9)-(-2)] might seem intimidating at first, but by systematically applying the order of operations, we can unravel its complexity and reveal its underlying simplicity. This process is akin to peeling back the layers of an onion, with each layer representing a step in the simplification process. So, let's delve into the first layer and begin our journey towards the solution.
1. Simplifying Parentheses: The Initial Steps
Our primary focus is on simplifying the expressions enclosed within parentheses in the given equation (5-3)x[(-6)+(5+1)-(-5)]-[(5+6)-(-7+9)-(-2)]. Starting with the leftmost set of parentheses, we encounter (5-3), which readily simplifies to 2. This seemingly small step is a significant stride towards unraveling the entire expression. Next, we address the parentheses (5+1), which yields 6. These initial simplifications pave the way for tackling the more complex expressions within the brackets. Remember, the order of operations dictates that we prioritize parentheses before any other operations. By addressing these inner expressions first, we reduce the overall complexity and make the subsequent steps more manageable. The act of simplifying parentheses is akin to clearing the underbrush in a forest, making it easier to navigate the terrain ahead. Each simplified expression becomes a building block for the next step, ultimately leading us to the final solution. The ability to quickly and accurately simplify parentheses is a crucial skill in mathematics. It not only saves time but also reduces the likelihood of errors. As we continue our journey through the expression (5-3)x[(-6)+(5+1)-(-5)]-[(5+6)-(-7+9)-(-2)], we'll see how this foundational skill plays a vital role in the overall solution.
2. Tackling Brackets: The Next Level of Simplification
Having simplified the innermost parentheses, we now turn our attention to the brackets in the expression (5-3)x[(-6)+(5+1)-(-5)]-[(5+6)-(-7+9)-(-2)]. The first set of brackets contains the expression [(-6)+(5+1)-(-5)]. We already know that (5+1) simplifies to 6, so we can substitute that into the expression, resulting in [-6 + 6 - (-5)]. Next, we simplify -6 + 6, which equals 0. This leaves us with [0 - (-5)]. Subtracting a negative number is equivalent to adding its positive counterpart, so 0 - (-5) becomes 0 + 5, which equals 5. Therefore, the first set of brackets simplifies to 5. Moving on to the second set of brackets, we have [(5+6)-(-7+9)-(-2)]. We first simplify the expressions within the parentheses: (5+6) equals 11, and (-7+9) equals 2. Substituting these values into the expression, we get [11 - 2 - (-2)]. Next, we perform the subtraction: 11 - 2 equals 9. This leaves us with [9 - (-2)]. Again, subtracting a negative number is the same as adding its positive counterpart, so 9 - (-2) becomes 9 + 2, which equals 11. Therefore, the second set of brackets simplifies to 11. By systematically working through each set of brackets, we have significantly reduced the complexity of the original expression (5-3)x[(-6)+(5+1)-(-5)]-[(5+6)-(-7+9)-(-2)]. This step-by-step approach allows us to maintain accuracy and avoid errors. The ability to simplify brackets is a crucial skill in mathematics, as it allows us to tackle more complex expressions with confidence. As we continue our journey towards the final solution, we'll see how this skill builds upon the previous steps and paves the way for the remaining operations.
3. Multiplication and Subtraction: The Final Steps
With the parentheses and brackets simplified, we can now focus on the remaining operations in the expression (5-3)x[(-6)+(5+1)-(-5)]-[(5+6)-(-7+9)-(-2)]. We have already established that (5-3) equals 2, the first set of brackets simplifies to 5, and the second set of brackets simplifies to 11. Substituting these values back into the original expression, we get 2 x 5 - 11. According to the order of operations, multiplication takes precedence over subtraction. Therefore, we first perform the multiplication: 2 x 5 equals 10. This leaves us with the simple expression 10 - 11. Finally, we perform the subtraction: 10 - 11 equals -1. Therefore, the final solution to the expression (5-3)x[(-6)+(5+1)-(-5)]-[(5+6)-(-7+9)-(-2)] is -1. This final step demonstrates the importance of following the order of operations. By performing the multiplication before the subtraction, we arrived at the correct answer. Had we performed the subtraction first, we would have obtained an incorrect result. The ability to perform multiplication and subtraction accurately is a fundamental skill in mathematics. It is essential for solving a wide range of problems, from simple arithmetic to complex algebraic equations. As we have seen in this example, these operations, when combined with the order of operations, form the foundation for solving numerical expressions.
Conclusion: The Power of Order and Simplification
In conclusion, the numerical expression (5-3)x[(-6)+(5+1)-(-5)]-[(5+6)-(-7+9)-(-2)], which initially appeared complex, has been successfully solved through a systematic, step-by-step approach. By adhering to the order of operations (PEMDAS) and simplifying each component individually, we arrived at the solution of -1. This exercise highlights the importance of understanding and applying mathematical principles to break down complex problems into manageable steps. The ability to simplify expressions is not just a mathematical skill; it's a valuable problem-solving skill that can be applied to various aspects of life. By learning to break down complex tasks into smaller, more manageable steps, we can overcome challenges and achieve our goals more effectively. The expression (5-3)x[(-6)+(5+1)-(-5)]-[(5+6)-(-7+9)-(-2)] served as an excellent example of how the order of operations and simplification techniques can be used to solve complex problems. The systematic approach not only ensures accuracy but also promotes a deeper understanding of the underlying mathematical concepts. This understanding, in turn, empowers us to tackle even more challenging problems with confidence. The journey of solving this expression was not just about arriving at the answer; it was about developing a logical and systematic approach to problem-solving. This approach, once mastered, can be applied to a wide range of challenges, both within and beyond the realm of mathematics. The skills honed in this exercise, such as attention to detail, patience, and the ability to break down complex tasks, are invaluable assets in any field. So, embrace the power of order and simplification, and you'll be well-equipped to conquer any mathematical challenge that comes your way. The final solution, -1, is not just a number; it's a testament to the power of systematic problem-solving and the beauty of mathematical logic. This exercise serves as a reminder that even the most daunting challenges can be overcome with the right approach and a solid understanding of the fundamentals.
