Simplifying (-6-√(36-20))/(2(1)) A Step-by-Step Guide

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Introduction

In the realm of mathematics, unraveling complex expressions is a fundamental skill. This article delves into a step-by-step breakdown of the expression (-6-√(36-20))/(2(1)), providing a clear and concise explanation for students and enthusiasts alike. We will explore the order of operations, simplification techniques, and ultimately, the final solution. The main focus of this comprehensive guide is to simplify the mathematical expression effectively. Our journey will start from understanding the basic components of the expression, moving on to applying the order of operations correctly, and finally arriving at the most simplified form of the result. This detailed explanation will not only help you in solving this particular problem but also in understanding the underlying principles that govern mathematical simplification. Throughout this article, emphasis will be given to the logical steps involved in the process, ensuring that the reader not only gets the answer but also comprehends the method of arriving at it. Understanding these mathematical principles and simplification techniques is crucial for anyone looking to excel in mathematics, whether in academic pursuits or practical applications. The ability to break down and solve complex problems is a valuable skill that extends far beyond the classroom, and this article is designed to help you develop this very skill. So, let's embark on this mathematical journey together and unravel the solution to the given expression in a way that is both informative and engaging.

Understanding the Expression: A Deep Dive into the Components

Before we begin the simplification process, it's crucial to understand each component of the expression. The expression (-6-√(36-20))/(2(1)) is composed of several elements: a square root, subtraction, division, and multiplication. Let’s dissect these elements one by one to get a clearer picture. First, we have the numerator, which is -6-√(36-20). This part of the expression involves a square root and subtraction. The square root symbol (√) indicates that we need to find a number that, when multiplied by itself, equals the number inside the root, which in this case is 36-20. This operation inside the square root needs to be performed before we can subtract the result from -6. Understanding the order of operations is paramount here. Next, we have the denominator, which is 2(1). This is a straightforward multiplication operation. Multiplying 2 by 1 is a basic arithmetic operation that we will perform to simplify the denominator. The entire expression is a fraction, with the numerator as (-6-√(36-20)) and the denominator as 2(1). This fraction indicates that we need to divide the result of the numerator by the result of the denominator. Understanding the relationship between these components is key to solving the expression correctly. By breaking down the expression into its constituent parts, we can approach the simplification process in a more organized and methodical manner. This step-by-step approach is not only beneficial for this particular problem but also for tackling any complex mathematical expression. Recognizing the different operations and their relationships is the first step towards mastering mathematical problem-solving. So, with a clear understanding of the components, we are now ready to move on to the next phase: applying the order of operations.

Applying the Order of Operations (PEMDAS/BODMAS)

The order of operations is a fundamental concept in mathematics, dictating the sequence in which operations must be performed to arrive at the correct solution. The acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) and BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) serve as handy mnemonics to remember this order. In our expression, (-6-√(36-20))/(2(1)), we have several operations: subtraction, a square root (which can be considered an exponent), division, and multiplication. According to PEMDAS/BODMAS, we must first address any operations within parentheses or brackets. In this case, we have (36-20) inside the square root, which is equivalent to a bracketed operation. So, the first step is to perform the subtraction 36 - 20, which equals 16. The expression now becomes (-6-√16)/(2(1)). Next, we address exponents or orders. Here, we have a square root, which is an exponent operation. The square root of 16 (√16) is 4, as 4 multiplied by itself equals 16. Our expression now looks like this: (-6-4)/(2(1)). Moving on, we have multiplication and division. In the denominator, we have 2(1), which is a multiplication operation. Multiplying 2 by 1 gives us 2. So, the expression is now (-6-4)/2. Finally, we address addition and subtraction. In the numerator, we have -6-4, which is a subtraction operation. Subtracting 4 from -6 gives us -10. The expression is now simplified to -10/2. The last step is to perform the division: -10 divided by 2, which equals -5. Therefore, by diligently following the order of operations, we have successfully simplified the expression to its final solution. This methodical approach is crucial for accuracy and consistency in mathematical problem-solving. Understanding and applying PEMDAS/BODMAS correctly ensures that we tackle complex expressions in the right sequence, avoiding errors and leading us to the correct answer. This principle is not just applicable to this specific problem but is a cornerstone of mathematical computation in general.

Step-by-Step Simplification of (-6-√(36-20))/(2(1))

Now, let's walk through the step-by-step simplification of the expression (-6-√(36-20))/(2(1)) to solidify our understanding. This process will illustrate how each operation is performed in the correct order, leading us to the final answer.

  1. Simplify inside the square root:

    • We start with the operation inside the square root: 36 - 20.
    • Performing the subtraction, we get 16.
    • The expression now becomes (-6-√16)/(2(1)).

  2. Evaluate the square root:

    • Next, we evaluate the square root of 16, which is √16.
    • Since 4 multiplied by 4 equals 16, √16 = 4.
    • The expression now transforms to (-6-4)/(2(1)).

  3. Simplify the numerator:

    • In the numerator, we have -6 - 4.
    • Performing the subtraction, we get -10.
    • The expression is now -10/(2(1)).

  4. Simplify the denominator:

    • In the denominator, we have 2(1), which is a multiplication operation.
    • Multiplying 2 by 1 gives us 2.
    • The expression is now -10/2.

  5. Perform the division:

    • Finally, we perform the division -10 / 2.
    • Dividing -10 by 2, we get -5.
    • Therefore, the simplified answer is -5.

By following these steps meticulously, we have successfully simplified the expression. Each step builds upon the previous one, ensuring that we adhere to the order of operations and arrive at the correct solution. This step-by-step approach not only helps in solving this particular problem but also provides a framework for tackling other complex mathematical expressions. The key is to break down the problem into manageable parts, address each part in the correct order, and meticulously perform the necessary operations. This methodical approach minimizes errors and enhances understanding, making mathematical problem-solving a more straightforward and rewarding endeavor. Now that we have seen the detailed step-by-step simplification, let's move on to discussing some common mistakes to avoid while solving such expressions.

Common Mistakes to Avoid

When simplifying mathematical expressions like (-6-√(36-20))/(2(1)), certain common mistakes can lead to incorrect answers. Being aware of these pitfalls can help you avoid them and ensure accuracy in your calculations. One of the most frequent errors is disregarding the order of operations (PEMDAS/BODMAS). For instance, someone might prematurely subtract 6 from the square root term without first evaluating the square root or the expression within it. This can lead to a completely different result. Another common mistake is incorrectly simplifying the square root. For example, not calculating 36-20 inside the square root first and directly taking the square root of 36 can lead to an erroneous simplification. It's crucial to remember that the operation inside the square root must be performed before taking the square root itself. Errors in basic arithmetic, such as incorrect subtraction or division, can also derail the simplification process. Even a small mistake in these fundamental operations can propagate through the rest of the calculation, leading to a wrong final answer. Another potential pitfall is misunderstanding the signs, especially when dealing with negative numbers. For example, incorrectly handling the subtraction of a number can change the sign of the result, leading to an incorrect solution. It’s important to pay close attention to the signs and their impact on the operations. Lastly, rushing through the steps without proper attention to detail can lead to errors. Simplifying complex expressions requires patience and a methodical approach. Skipping steps or performing calculations mentally without writing them down can increase the likelihood of mistakes. By being mindful of these common mistakes and taking a careful, step-by-step approach, you can significantly improve your accuracy and confidence in solving mathematical expressions. The key is to understand the underlying principles, apply the order of operations correctly, and pay close attention to the details of each step.

Conclusion: Mastering Mathematical Simplification

In conclusion, mastering the simplification of mathematical expressions like (-6-√(36-20))/(2(1)) is a crucial skill in mathematics. Throughout this article, we have dissected the expression, broken it down into manageable parts, and systematically simplified it step by step. We began by understanding the components of the expression, identifying the various operations involved, and recognizing the importance of the order of operations. We then meticulously applied the PEMDAS/BODMAS rule, ensuring that each operation was performed in the correct sequence. This methodical approach not only led us to the correct solution but also provided a clear framework for tackling other complex expressions. By simplifying inside the square root first, then evaluating the square root, simplifying the numerator and denominator, and finally performing the division, we arrived at the final answer of -5. We also highlighted common mistakes to avoid, such as disregarding the order of operations, incorrectly simplifying the square root, making errors in basic arithmetic, misunderstanding the signs, and rushing through the steps. Awareness of these pitfalls can significantly improve accuracy and prevent errors in calculations. The ability to simplify mathematical expressions is not just about arriving at the correct answer; it's about developing a logical and methodical approach to problem-solving. This skill is invaluable not only in academic settings but also in various real-world applications. By understanding the underlying principles, applying the correct techniques, and practicing regularly, anyone can master mathematical simplification and confidently tackle complex problems. So, embrace the challenge, hone your skills, and unlock the power of mathematical simplification.