Solving $-7\{5-[(-10+2) \div 4-(-7)]-(5-8)+8\}=$ A Step-by-Step Guide

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Understanding the Order of Operations

When dealing with complex mathematical expressions like −7{5−[(−10+2)÷4−(−7)]−(5−8)+8}=-7\{5-[(-10+2) \div 4-(-7)]-(5-8)+8\}=, it's crucial to follow the correct 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). By adhering to this order, we can systematically simplify the expression and arrive at the correct answer. In our specific case, we'll start by addressing the innermost parentheses and work our way outwards, performing division and subtraction as we go, before finally applying the multiplication and outer operations.

Step-by-Step Solution

Let's break down the expression −7{5−[(−10+2)÷4−(−7)]−(5−8)+8}=-7\{5-[(-10+2) \div 4-(-7)]-(5-8)+8\}= step by step:

  1. Innermost Parentheses: Start by simplifying the expressions within the innermost parentheses. We have (−10+2)(-10+2) and (5−8)(5-8).

    • (−10+2)=−8(-10 + 2) = -8
    • (5−8)=−3(5 - 8) = -3

    Now our expression looks like this: −7{5−[(−8)÷4−(−7)]−(−3)+8}=-7\{5-[(-8) \div 4-(-7)]-(-3)+8\}=

  2. Division: Next, perform the division operation within the brackets. We have (−8)÷4(-8) \div 4.

    • (−8)÷4=−2(-8) \div 4 = -2

    The expression now becomes: −7{5−[−2−(−7)]−(−3)+8}=-7\{5-[-2-(-7)]-(-3)+8\}=

  3. Subtraction within Brackets: Now, handle the subtraction within the square brackets. We have −[−2−(−7)]-[-2-(-7)].

    • −2−(−7)=−2+7=5-2 - (-7) = -2 + 7 = 5
    • So, −[−2−(−7)]=−5-[-2-(-7)] = -5

    Our expression is now: −7{5−5−(−3)+8}=-7\{5 - 5 - (-3) + 8\}=

  4. Subtraction and Addition within Curly Braces: Next, we perform the subtraction and addition within the curly braces.

    • 5−5−(−3)+8=5−5+3+8=115 - 5 - (-3) + 8 = 5 - 5 + 3 + 8 = 11

    The expression simplifies to: −7×11=-7 \times 11=

  5. Final Multiplication: Finally, multiply the result by -7.

    • −7×11=−77-7 \times 11 = -77

Detailed Breakdown of Each Operation

To further clarify the process, let's delve deeper into each operation performed:

  • Innermost Parentheses Simplification: The initial simplification of (−10+2)(-10 + 2) to −8-8 and (5−8)(5 - 8) to −3-3 is a straightforward application of addition and subtraction rules with integers. When adding a negative number, you effectively subtract its absolute value. Subtracting a larger number from a smaller one results in a negative value, thus simplifying these expressions is the crucial first step in following the PEMDAS order of operations.
  • Division Operation: Dividing −8-8 by 44 gives us −2-2. This is a basic division operation, ensuring we handle the signs correctly. A negative number divided by a positive number yields a negative result. Performing this division early in the process keeps the expression manageable and prevents potential errors later on. This step exemplifies how crucial division is within the context of the equation and the order of operations.
  • Subtraction within Brackets: The subtraction −[−2−(−7)]-[-2 - (-7)] requires careful attention to signs. Subtracting a negative number is equivalent to adding its positive counterpart. Therefore, −2−(−7)-2 - (-7) becomes −2+7-2 + 7, which equals 55. Then, applying the negative sign outside the bracket gives us −5-5. This step showcases the importance of accurately managing negative signs and their impact on the final result.
  • Simplifying Curly Braces: Inside the curly braces, we have 5−5−(−3)+85 - 5 - (-3) + 8. Simplifying this involves combining addition and subtraction. The expression becomes 0+3+80 + 3 + 8, which equals 1111. This simplification process reduces the complexity of the equation, making it easier to handle the final multiplication. This highlights how reducing complexity is a key part of solving complex mathematical expressions.
  • Final Multiplication: The final step involves multiplying −7-7 by the simplified result, 1111. This straightforward multiplication gives us −77-77. The negative sign is critical here, as it reflects the combined effect of the negative coefficient and the simplified expression. The correct management of signs in this last step ensures the accurate calculation of the final answer.

Common Mistakes to Avoid

When solving complex expressions, it’s easy to make mistakes if you're not careful. Here are some common pitfalls to watch out for:

  • Ignoring the Order of Operations: One of the biggest mistakes is not following PEMDAS. Always address parentheses, exponents, multiplication and division, and then addition and subtraction in the correct order.
  • Sign Errors: Negative signs can be tricky. Make sure to handle them carefully, especially when subtracting negative numbers.
  • Incorrect Distribution: If there are any distribution operations, ensure they are performed correctly before moving on to other steps.
  • Rushing Through the Steps: Take your time and solve the expression step by step. Rushing can lead to errors.

Conclusion

By following the order of operations meticulously and paying close attention to detail, we successfully simplified the expression −7{5−[(−10+2)÷4−(−7)]−(5−8)+8}=-7\{5-[(-10+2) \div 4-(-7)]-(5-8)+8\}= to -77. This detailed walkthrough underscores the significance of adhering to established mathematical principles and employing a systematic approach to complex problems. Each step, from handling parentheses to managing signs, plays a crucial role in achieving accuracy. Understanding and applying these concepts not only aids in solving equations but also enhances overall mathematical proficiency. The final answer, -77, is a testament to the precision and care exercised throughout the solution process. This comprehensive method can be applied to a multitude of complex calculations, reinforcing the importance of a structured approach in mathematics.