Evaluate The Expression 8 + 8 ÷ (13 - (-20) ÷ (-4))
In this article, we will delve into the process of evaluating the mathematical expression:
8 + 8 ÷ (13 - (-20) ÷ (-4))
This expression involves a combination of arithmetic operations, including addition, division, subtraction, and negative numbers. To arrive at the correct answer, we must follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This guide will provide a comprehensive, step-by-step solution, ensuring clarity and understanding for anyone looking to improve their arithmetic skills. This meticulous approach not only helps in solving this specific problem but also builds a strong foundation for tackling more complex mathematical challenges. Let's break down each step with detailed explanations to ensure clarity and accuracy in our calculation. Understanding the order of operations is crucial, and this detailed walkthrough will solidify your grasp on this essential mathematical principle.
Understanding the Order of Operations (PEMDAS/BODMAS)
Before we begin, let's reiterate the order of operations, which is the backbone of solving any mathematical expression. PEMDAS stands for:
- Parentheses (or Brackets)
- Exponents (or Orders)
- Multiplication and Division (from left to right)
- Addition and Subtraction (from left to right)
Alternatively, some use the acronym BODMAS, where B stands for Brackets, O for Orders, D for Division, M for Multiplication, A for Addition, and S for Subtraction. Both acronyms represent the same hierarchy of operations. Adhering to this order is paramount to ensure accurate results. In our expression, we have parentheses, division, addition, and subtraction, each of which must be addressed in the correct sequence. Neglecting this order can lead to incorrect answers, emphasizing the importance of a methodical approach. By following PEMDAS/BODMAS, we can systematically simplify the expression, making it easier to manage and reducing the likelihood of errors. This principle not only applies to this specific problem but is a universal rule in mathematics, essential for consistent and correct calculations across various mathematical contexts.
Step 1: Simplify within the Parentheses
Our expression is:
8 + 8 ÷ (13 - (-20) ÷ (-4))
The first step, according to PEMDAS, is to deal with the parentheses. Inside the parentheses, we have:
13 - (-20) ÷ (-4)
Within these parentheses, we have subtraction and division. According to the order of operations, division takes precedence. So, we first perform the division:
(-20) ÷ (-4)
When dividing a negative number by a negative number, the result is positive. So:
(-20) ÷ (-4) = 5
Now, we substitute this back into the parentheses:
13 - 5
Performing the subtraction:
13 - 5 = 8
So, the expression within the parentheses simplifies to 8. This step is critical as it reduces the complexity of the overall expression, making it easier to manage in subsequent steps. By focusing on the innermost operations first, we avoid confusion and ensure accuracy. This methodical approach of simplifying within parentheses is a fundamental technique in mathematical problem-solving, applicable across various types of expressions and equations. It is a clear demonstration of how breaking down a complex problem into smaller, manageable steps can lead to a straightforward solution. Let's continue with the next steps to fully evaluate the expression.
Step 2: Substitute the Simplified Parenthetical Expression
Now that we've simplified the expression within the parentheses, we substitute the result back into the original expression. Our original expression was:
8 + 8 ÷ (13 - (-20) ÷ (-4))
And we found that:
(13 - (-20) ÷ (-4)) = 8
Substituting this result, our expression now becomes:
8 + 8 ÷ 8
This simplified form is much easier to work with. By replacing the complex expression inside the parentheses with its simplified value, we've reduced the expression to a more manageable state. This substitution step is a key technique in solving mathematical problems, as it allows us to break down complex equations into simpler parts. It is also a clear illustration of the power of the order of operations, as correctly simplifying the parentheses first paves the way for the remaining calculations. We are now one step closer to the final answer. The next step will focus on addressing the remaining operations, ensuring we continue to follow the correct order and arrive at the accurate solution. Let's proceed with the next step to further simplify the expression.
Step 3: Perform the Division
After simplifying the expression within the parentheses and substituting it back, we now have:
8 + 8 ÷ 8
According to PEMDAS, division must be performed before addition. So, we focus on the division operation:
8 ÷ 8
This is a straightforward division:
8 ÷ 8 = 1
Now, we substitute this result back into the expression:
8 + 1
This step is crucial because adhering to the order of operations is vital for accurate calculations. Performing the division before addition ensures we follow the correct mathematical procedure. The simplicity of this step highlights how breaking down a complex expression into smaller, manageable operations can make the problem much easier to solve. We are now left with a simple addition problem, which will lead us to the final answer. This stage of the solution underscores the importance of methodical calculation and the benefits of following the established rules of arithmetic. Let's complete the final step to find the solution.
Step 4: Perform the Addition
We've simplified the expression to:
8 + 1
Now, we perform the final addition operation:
8 + 1 = 9
Therefore, the final result of the expression is 9. This final step is the culmination of our step-by-step approach, demonstrating how each individual operation contributes to the overall solution. The simplicity of the addition at this stage underscores the effectiveness of following the order of operations and breaking down the problem into manageable parts. Achieving this result validates our methodical approach and confirms the accuracy of our calculations. This comprehensive walkthrough not only solves the problem but also reinforces the importance of precision and adherence to mathematical principles in problem-solving. Let's summarize our solution in the conclusion.
Conclusion
By following the order of operations (PEMDAS), we have successfully evaluated the expression:
8 + 8 ÷ (13 - (-20) ÷ (-4))
Our step-by-step solution is as follows:
- Simplify within the parentheses:
(13 - (-20) ÷ (-4)) = 8 - Substitute the simplified parenthetical expression:
8 + 8 ÷ 8 - Perform the division:
8 ÷ 8 = 1 - Perform the addition:
8 + 1 = 9
Therefore, the final answer is:
9
This exercise demonstrates the importance of following the correct order of operations in mathematical calculations. Each step, from simplifying within parentheses to performing the final addition, played a crucial role in arriving at the correct answer. This methodical approach not only solves the problem but also provides a clear and understandable process that can be applied to other mathematical expressions. The ability to break down complex problems into smaller, manageable steps is a valuable skill in mathematics and beyond. This detailed solution serves as a guide for anyone looking to improve their arithmetic skills and reinforces the significance of precision and accuracy in mathematical problem-solving.