Solving The Complex Expression $2[\{8-[3-(-2+\{5-(-1)\}))]\} +1]-6$ A Step-by-Step Guide

Introduction to Order of Operations

In mathematics, we often encounter complex expressions that require us to perform multiple operations. To ensure that we arrive at the correct answer, we need to follow a specific order of operations. This order is commonly remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Adhering to PEMDAS is crucial for simplifying mathematical expressions accurately.

The expression $2[\{8-[3-(-2+\{5-(-1)\}))]\} +1]-6$ provides an excellent opportunity to apply the order of operations. This expression involves nested parentheses, brackets, and braces, making it essential to proceed step by step. By carefully following the order of operations, we can break down this complex expression into smaller, manageable parts and ultimately arrive at the correct solution. This process not only helps in solving the problem but also enhances our understanding of mathematical principles and problem-solving strategies.

Breaking Down the Expression: A Step-by-Step Approach

To simplify the expression $2[\{8-[3-(-2+\{5-(-1)\}))]\} +1]-6$, we will meticulously follow the order of operations. This involves addressing the innermost parentheses first and gradually working our way outwards. Each step will be carefully executed to ensure accuracy and clarity. The goal is to transform the complex expression into a simplified numerical value by systematically applying the rules of arithmetic.

Our first step involves dealing with the innermost parentheses: (5 - (-1)). Understanding how to handle negative numbers is crucial here. Subtracting a negative number is equivalent to adding its positive counterpart. This principle is fundamental in simplifying expressions involving negative signs. By correctly applying this rule, we can simplify the innermost parentheses and move on to the next level of complexity in the expression.

This step-by-step approach not only helps in solving the problem but also builds a strong foundation in mathematical problem-solving. By understanding each operation and its impact on the expression, we can develop a deeper appreciation for the elegance and precision of mathematics. The following sections will detail each step of the simplification process, providing a clear and comprehensive guide to solving the expression.

Step-by-Step Solution

1. Simplify the Innermost Parentheses: (5 - (-1))

Our journey to solve the mathematical expression $2[\{8-[3-(-2+\{5-(-1)\}))]\} +1]-6$ begins with the innermost parentheses. The expression inside these parentheses is (5 - (-1)). The key to simplifying this is understanding how to handle subtraction of a negative number. Subtracting a negative number is the same as adding its positive counterpart. Therefore, 5 - (-1) becomes 5 + 1.

Performing this addition, we get 5 + 1 = 6. This simplifies the innermost part of the expression, allowing us to rewrite the original expression with this simplification. Substituting 6 for (5 - (-1)), the expression now looks like this: $2[\{8-[3-(-2+\{6\}))]\} +1]-6$. This step is crucial as it reduces the complexity of the expression and allows us to proceed with the next level of operations.

Understanding this basic principle of arithmetic is essential for tackling more complex mathematical problems. It forms the foundation for handling expressions involving negative numbers and sets the stage for the subsequent steps in simplifying the given expression. Each simplification brings us closer to the final solution, highlighting the importance of methodical and accurate application of mathematical rules.

2. Simplify the Next Parentheses: (-2 + 6)

Having simplified the innermost parentheses, we now focus on the next set of parentheses in the expression $2[\{8-[3-(-2+\{6\}))]\} +1]-6$. These parentheses contain the expression (-2 + 6). This step involves adding a negative number to a positive number. To solve this, we consider the difference between the absolute values of the numbers and use the sign of the number with the larger absolute value.

In this case, we have -2 + 6. The absolute value of -2 is 2, and the absolute value of 6 is 6. The difference between 6 and 2 is 4. Since 6 has a larger absolute value and is positive, the result of -2 + 6 is 4. Substituting this back into our expression, we now have $2[\{8-[3-(4)]\} +1]-6$.

This step demonstrates the importance of understanding how to add numbers with different signs. It's a fundamental concept in arithmetic and is crucial for simplifying more complex expressions. By carefully applying this rule, we continue to reduce the complexity of the original expression, bringing us closer to the final answer. Each step builds upon the previous one, emphasizing the sequential nature of mathematical problem-solving.

3. Simplify the Parentheses: (3 - (4))

Moving forward in our simplification journey, we address the parentheses containing the expression (3 - (4)) within $2[\{8-[3-(4)]\} +1]-6$. Here, we are subtracting a larger number from a smaller number, which will result in a negative value. This step requires a clear understanding of how to handle subtraction in such cases.

To simplify 3 - 4, we perform the subtraction, which gives us -1. Substituting this result back into the expression, we now have $2[\{8-[-1]\} +1]-6$. This simplification is crucial as it further reduces the complexity of the expression and allows us to proceed with the next set of operations.

Understanding the rules of subtraction, especially when dealing with positive and negative numbers, is essential for accurate mathematical calculations. This step highlights the importance of paying close attention to the signs of numbers and how they affect the outcome of the operation. By mastering these fundamental concepts, we can confidently tackle more challenging mathematical problems.

4. Simplify the Brackets: [8 - (-1)]

Now, we turn our attention to the brackets in the expression $2[\{8-[-1]\} +1]-6$. Inside the brackets, we have [8 - (-1)]. Similar to our earlier step with parentheses, we are again subtracting a negative number. Subtracting a negative number is equivalent to adding its positive counterpart. Therefore, 8 - (-1) becomes 8 + 1.

Performing this addition, we get 8 + 1 = 9. Substituting this back into the expression, we now have $2[\{9\} +1]-6$. This simplification significantly reduces the complexity of the expression, bringing us closer to the final solution. It also reinforces the principle of how subtracting a negative number transforms into addition, a crucial concept in arithmetic.

This step demonstrates the importance of recognizing and applying fundamental mathematical rules consistently. By correctly handling the subtraction of a negative number, we have simplified a significant portion of the expression. Each step taken contributes to a clearer and more manageable form of the original problem, illustrating the power of step-by-step simplification in mathematics.

5. Simplify the Braces: {9} + 1

We continue our simplification process by focusing on the braces in the expression $2[\{9\} +1]-6$. Inside the braces, we have {9} + 1. This step involves adding 1 to 9, which is a straightforward addition operation.

Performing the addition, we get 9 + 1 = 10. Substituting this result back into the expression, we now have $2[10]-6$. This simplification is a significant step towards solving the expression, as it eliminates the braces and further reduces the complexity of the problem. The expression is now much simpler and easier to manage.

This step reinforces the basic addition operation and its role in simplifying mathematical expressions. By accurately performing this addition, we have moved closer to the final answer. Each step in the simplification process builds upon the previous one, highlighting the importance of a systematic and methodical approach to solving mathematical problems.

6. Perform the Multiplication: 2[10]

With the expression now simplified to $2[10]-6$, we focus on the multiplication operation. The notation 2[10] indicates that we need to multiply 2 by 10. This is a fundamental arithmetic operation that is essential for solving the expression.

Performing the multiplication, we get 2 * 10 = 20. Substituting this result back into the expression, we now have 20 - 6. This simplification is a crucial step as it further reduces the complexity of the expression and sets the stage for the final subtraction operation. The expression is now in a simple form that is easy to solve.

This step highlights the importance of understanding the order of operations, where multiplication takes precedence over addition and subtraction. By correctly performing the multiplication, we have moved closer to the final solution. Each step in the simplification process brings us closer to the answer, demonstrating the power of methodical and accurate application of mathematical rules.

7. Perform the Final Subtraction: 20 - 6

Finally, we arrive at the last step in simplifying the expression 20 - 6. This step involves a simple subtraction operation. Subtracting 6 from 20 will give us the final answer to the expression. This is the culmination of all the previous steps, where we have systematically reduced the complexity of the original expression.

Performing the subtraction, we get 20 - 6 = 14. This is the final result of our calculation. Therefore, the value of the expression $2[\{8-[3-(-2+\{5-(-1)\}))]\} +1]-6$ is 14. This result is the outcome of carefully applying the order of operations and accurately performing each arithmetic operation.

This final step underscores the importance of precision and attention to detail in mathematical problem-solving. By correctly executing the subtraction, we have arrived at the solution. The entire process, from the initial complex expression to the final answer, demonstrates the power of a methodical and step-by-step approach to simplifying mathematical problems.

Conclusion: The Final Result

In conclusion, by meticulously following the order of operations and performing each step with precision, we have successfully simplified the complex mathematical expression $2[\{8-[3-(-2+\{5-(-1)\}))]\} +1]-6$. The final result of this step-by-step simplification is 14. This outcome is a testament to the importance of adhering to mathematical rules and the power of systematic problem-solving.

The process of breaking down the expression into smaller, manageable parts allowed us to tackle each operation individually. From simplifying the innermost parentheses to performing the final subtraction, each step contributed to the overall solution. This approach not only helped in solving the problem but also reinforced the fundamental principles of arithmetic and the significance of order of operations.

Understanding and applying these principles is crucial for success in mathematics. The ability to simplify complex expressions and arrive at the correct answer is a valuable skill that can be applied in various fields. By mastering these skills, individuals can confidently approach mathematical challenges and excel in their academic and professional pursuits. The solution to this expression serves as a practical example of how mathematical concepts can be applied to solve real-world problems.