Simplifying Numerical Expressions: A Step-by-Step Guide

In this article, we'll break down the process of simplifying a complex numerical expression. We'll use the order of operations (PEMDAS/BODMAS) and go through each step to make it easy to understand. Our example expression is: $-11+7-2\{-4+1-[-2(-3+4)-2+4+7-8]-4\}$. Let's dive in!

Understanding the Order of Operations

Before we begin, it's crucial to understand the order of operations, often remembered by the acronyms PEMDAS or BODMAS:

• Parentheses / Brackets
• Exponents / Orders
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

Following this order ensures we simplify expressions correctly and consistently. Numerical expressions, especially those involving nested brackets and multiple operations, can seem daunting at first glance. The key to tackling these problems lies in a systematic approach. By meticulously following the order of operations (PEMDAS/BODMAS), we can break down even the most complex expressions into manageable steps. This involves starting with the innermost brackets or parentheses and working our way outwards. As we simplify each part, we must pay close attention to the signs and ensure that each operation is performed in the correct sequence. With practice and a keen eye for detail, simplifying numerical expressions becomes not only easier but also a valuable skill in mathematics and beyond. Remember, each step builds upon the previous one, so accuracy at each stage is paramount to achieving the correct final answer. The more you practice, the more confident and proficient you will become in handling these types of problems. Keep in mind that the goal is not just to arrive at the correct answer, but also to understand the underlying principles and logic that govern the process. This understanding will empower you to tackle even more challenging mathematical problems in the future. So, let's embark on this journey of simplification, one step at a time, with patience and a determination to master the art of numerical expressions.

Step-by-Step Simplification

Let's simplify the expression step-by-step:

$-11+7-2\{-4+1-[-2(-3+4)-2+4+7-8]-4\}$

1. Innermost Parentheses: Start with the innermost parentheses: $(-3+4) = 1$.

   $-11+7-2\{-4+1-[-2(1)-2+4+7-8]-4\}$

2. Multiplication inside the Brackets: Next, perform the multiplication inside the square brackets: $-2(1) = -2$.

   $-11+7-2\{-4+1-[-2-2+4+7-8]-4\}$

3. Simplifying inside the Square Brackets: Now, simplify the expression inside the square brackets: $-2-2+4+7-8 = -1$.

   $-11+7-2\{-4+1-[-1]-4\}$

4. Removing the Square Brackets: Remove the square brackets. Note that $-[-1] = +1$.

   $-11+7-2\{-4+1+1-4\}$

5. Simplifying inside the Curly Braces: Simplify the expression inside the curly braces: $-4+1+1-4 = -6$.

   $-11+7-2\{-6\}$

6. Multiplication: Perform the multiplication: $-2\{-6\} = 12$.

   $-11+7+12$

7. Addition and Subtraction: Finally, perform the addition and subtraction from left to right: $-11+7+12 = -4+12 = 8$.

Therefore, the simplified expression is $8$.

Detailed Breakdown of Each Step

Step 1: Innermost Parentheses

Our initial expression is $-11+7-2\{-4+1-[-2(-3+4)-2+4+7-8]-4\}$. The innermost parentheses contain the expression $(-3+4)$. This is a simple addition operation. By adding $-3$ and $4$, we get $1$. So, $(-3+4) = 1$. Substituting this back into the original expression, we have: $-11+7-2\{-4+1-[-2(1)-2+4+7-8]-4\}$. This step is crucial as it simplifies the innermost part of the expression, allowing us to move outwards and tackle the more complex components. Always ensure the signs are correct and the arithmetic is accurate to avoid errors in subsequent steps. This methodical approach ensures a clear and straightforward path to simplifying the entire expression. Remember, the goal is to break down the problem into smaller, more manageable pieces, making it easier to solve.

Step 2: Multiplication inside the Brackets

After simplifying the innermost parentheses, our expression is now $-11+7-2\{-4+1-[-2(1)-2+4+7-8]-4\}$. Inside the square brackets, we have a multiplication operation: $-2(1)$. Multiplying $-2$ by $1$ gives us $-2$. Replacing $-2(1)$ with $-2$ in the expression, we get: $-11+7-2\{-4+1-[-2-2+4+7-8]-4\}$. This multiplication step is vital as it further simplifies the terms within the brackets. Correctly performing this operation ensures the accuracy of the subsequent steps. Pay close attention to the negative signs to avoid common errors. This methodical approach of simplifying step-by-step allows us to manage the complexity of the expression more effectively. Remember, accuracy at each step is crucial for arriving at the correct final answer. The more you practice these steps, the more comfortable and confident you will become in handling similar expressions. Each simplification brings us closer to the final solution, making the entire process more manageable and less daunting.

Step 3: Simplifying inside the Square Brackets

Our expression now looks like this: $-11+7-2\{-4+1-[-2-2+4+7-8]-4\}$. We need to simplify the expression inside the square brackets: $-2-2+4+7-8$. Combining these numbers: $-2 - 2 = -4$, then $-4 + 4 = 0$, then $0 + 7 = 7$, and finally $7 - 8 = -1$. So, the simplified expression inside the square brackets is $-1$. Substituting this back into the main expression, we get: $-11+7-2\{-4+1-[-1]-4\}$. This step is critical as it consolidates multiple terms into a single value, reducing the complexity of the expression. Ensure each addition and subtraction is performed accurately to avoid errors. This meticulous approach helps maintain clarity and accuracy throughout the simplification process. Remember, breaking down the problem into smaller, manageable steps makes it easier to solve and minimizes the chances of making mistakes. By carefully simplifying each part, we move closer to the final solution with confidence.

Step 4: Removing the Square Brackets

Following the simplification inside the square brackets, our expression is $-11+7-2\{-4+1-[-1]-4\}$. Now, we need to remove the square brackets. Notice that we have $-[-1]$, which simplifies to $+1$. Replacing $-[-1]$ with $+1$, our expression becomes: $-11+7-2\{-4+1+1-4\}$. Removing brackets often involves paying close attention to signs, especially when dealing with negative numbers. This step simplifies the expression by eliminating a layer of grouping, making it easier to proceed with the next operations. Accuracy in this step is crucial to maintain the integrity of the expression and avoid errors. Remember, each step is a building block that contributes to the final solution. By carefully handling the signs and performing the necessary operations, we continue to move closer to the simplified form of the expression.

Step 5: Simplifying inside the Curly Braces

After removing the square brackets, our expression is now $-11+7-2\{-4+1+1-4\}$. We need to simplify the expression inside the curly braces: $-4+1+1-4$. Combining these terms: $-4 + 1 = -3$, then $-3 + 1 = -2$, and finally $-2 - 4 = -6$. So, the simplified expression inside the curly braces is $-6$. Substituting this back into the main expression, we get: $-11+7-2\{-6\}$. This step is essential as it further reduces the complexity of the expression by consolidating multiple terms into a single value. Accuracy is key to ensure the correct final result. Remember, the order of operations guides us in simplifying the expression systematically. By carefully performing each addition and subtraction, we continue to move closer to the final simplified form.

Step 6: Multiplication

Following the simplification inside the curly braces, our expression is $-11+7-2\{-6\}$. Now, we perform the multiplication: $-2\{-6\}$. Multiplying $-2$ by $-6$ gives us $12$. Therefore, our expression becomes: $-11+7+12$. This multiplication step is crucial as it eliminates the curly braces and combines the constant term with the rest of the expression. Paying close attention to the signs is essential to avoid errors. In this case, multiplying two negative numbers results in a positive number. This step simplifies the expression further, bringing us closer to the final solution. Remember, each step in the simplification process builds upon the previous one, and accuracy at each stage is vital for achieving the correct final result.

Step 7: Addition and Subtraction

After performing the multiplication, our expression is $-11+7+12$. Now, we perform the addition and subtraction from left to right. First, we add $-11$ and $7$, which gives us $-4$. So, the expression becomes $-4+12$. Next, we add $-4$ and $12$, which gives us $8$. Therefore, the simplified expression is $8$. This final step combines all the remaining terms to arrive at the solution. Accuracy in performing the addition and subtraction is crucial for obtaining the correct final answer. This methodical approach, following the order of operations, ensures that we have simplified the expression correctly and efficiently. Remember, the key to simplifying complex expressions lies in breaking them down into smaller, manageable steps and carefully performing each operation. This step-by-step approach minimizes the chances of errors and leads us to the correct solution.

Conclusion

By following the order of operations (PEMDAS/BODMAS) and breaking down the expression into smaller steps, we successfully simplified the expression $-11+7-2\{-4+1-[-2(-3+4)-2+4+7-8]-4\}$ to $8$. Remember to always start with the innermost parentheses and work your way outwards, paying close attention to signs and arithmetic along the way. Practice makes perfect, so keep working on similar problems to improve your skills!

For more information on the order of operations, you can visit Khan Academy's article on Order of Operations.