Evaluating 7 × [(5 - 2) - 10 + (6 × 4 - 4)] A Step-by-Step Guide

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Evaluating mathematical expressions can sometimes feel like navigating a maze, but with a step-by-step approach and a clear understanding of the order of operations, even complex expressions can be simplified with ease. In this article, we will delve into the process of evaluating the expression 7 × [(5 - 2) - 10 + (6 × 4 - 4)], breaking it down into manageable steps to arrive at the final answer. By meticulously following the order of operations, also known as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), we will unravel the layers of this expression and uncover its value. This step-by-step guide is designed to enhance your understanding of mathematical operations and empower you to tackle similar problems with confidence.

Understanding the Order of Operations (PEMDAS/BODMAS)

Before we dive into the expression itself, it's crucial to understand the order of operations. This set of rules dictates the sequence in which mathematical operations should be performed to ensure accurate results. The acronym PEMDAS (or BODMAS in some regions) serves as a handy mnemonic:

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

By adhering to this order, we can systematically simplify the expression, ensuring that each operation is performed at the correct stage. Neglecting the order of operations can lead to incorrect answers, emphasizing the importance of mastering this fundamental concept. This principle is the cornerstone of accurate mathematical calculations and is essential for success in algebra and beyond. In this detailed breakdown, we'll see how PEMDAS guides us through each step, ensuring we arrive at the correct solution.

Step-by-Step Evaluation of the Expression

Now, let's embark on the journey of evaluating the expression 7 × [(5 - 2) - 10 + (6 × 4 - 4)]. We will proceed step-by-step, meticulously applying the order of operations to unravel the expression and determine its value.

Step 1: Simplify within the Innermost Parentheses

Our first task is to address the expressions within the innermost parentheses. Looking at the expression, we find two sets of parentheses: (5 - 2) and (6 × 4 - 4). We'll tackle each of these individually.

  • (5 - 2) = 3

This is a straightforward subtraction operation. Subtracting 2 from 5 yields 3. This simplified value will be used in subsequent steps. This seemingly simple step is crucial as it forms the building block for further calculations.

  • (6 × 4 - 4)

Here, we have a combination of multiplication and subtraction within the parentheses. According to PEMDAS, we must perform the multiplication before the subtraction.

*   **6 × 4 = 24**

Multiplying 6 by 4 gives us 24. Now we can substitute this result back into the expression within the parentheses.

*   **24 - 4 = 20**

Subtracting 4 from 24 leaves us with 20. This completes the simplification of the second set of innermost parentheses. This step demonstrates the importance of prioritizing multiplication over subtraction within parentheses, a key aspect of PEMDAS.

After completing this first step, the expression now looks like this:

7 × [3 - 10 + 20]

We have successfully simplified the innermost parentheses, paving the way for the next step in the evaluation process. The careful attention to detail in this initial step sets the foundation for an accurate final answer.

Step 2: Simplify within the Brackets

With the innermost parentheses simplified, we now turn our attention to the expression within the brackets: [3 - 10 + 20]. Inside the brackets, we have a combination of subtraction and addition. According to PEMDAS, addition and subtraction are performed from left to right.

  • 3 - 10 = -7

Subtracting 10 from 3 results in -7. This negative value is important to carry forward accurately. Dealing with negative numbers is a common aspect of mathematical expressions, and this step reinforces the importance of handling them correctly.

  • -7 + 20 = 13

Adding 20 to -7 gives us 13. This completes the simplification of the expression within the brackets. This step highlights the sequential nature of addition and subtraction when they appear together in an expression.

After simplifying the brackets, the expression now looks much simpler:

7 × 13

We have successfully reduced the expression within the brackets to a single value, making the final step straightforward. The methodical approach of PEMDAS has guided us through the complexities of the expression, leading us closer to the solution.

Step 3: Perform the Final Multiplication

We have now reached the final step in our evaluation. The expression has been simplified to 7 × 13. This is a simple multiplication operation.

  • 7 × 13 = 91

Multiplying 7 by 13 gives us 91. This is the final value of the expression. This final multiplication brings together all the previous steps, culminating in the solution. The accuracy of this final step depends on the correctness of all the preceding steps.

The Final Result

Therefore, the value of the expression 7 × [(5 - 2) - 10 + (6 × 4 - 4)] is 91. We have successfully evaluated the expression by meticulously following the order of operations, breaking it down into manageable steps, and performing each operation with precision. This result underscores the importance of adhering to PEMDAS and the power of a systematic approach in solving mathematical problems. The journey from the initial complex expression to the final simple answer showcases the elegance of mathematical principles.

Key Takeaways

Evaluating mathematical expressions requires a systematic approach, and the order of operations (PEMDAS/BODMAS) provides the framework for this approach. By adhering to this order, we can ensure that complex expressions are simplified accurately. Let's recap the key takeaways from this evaluation:

  • Prioritize Parentheses: Always simplify expressions within parentheses (or brackets) first, starting with the innermost sets.
  • Multiplication and Division Before Addition and Subtraction: Perform multiplication and division operations before addition and subtraction. When these operations appear together, work from left to right.
  • Addition and Subtraction from Left to Right: When only addition and subtraction remain, perform the operations from left to right.
  • Step-by-Step Approach: Break down complex expressions into smaller, manageable steps. This reduces the likelihood of errors and makes the process easier to follow.
  • Accuracy is Key: Pay close attention to each operation and ensure accuracy at every step. A small error early in the process can lead to a significantly incorrect final answer.

By internalizing these key takeaways, you can confidently tackle a wide range of mathematical expressions, ensuring accuracy and efficiency in your calculations. The ability to evaluate expressions is a fundamental skill in mathematics, and mastering this skill opens doors to more advanced concepts and problem-solving techniques. This methodical approach not only helps in finding the correct answer but also fosters a deeper understanding of mathematical principles.

Practice Makes Perfect

Evaluating mathematical expressions is a skill that improves with practice. The more you work through different types of expressions, the more comfortable and confident you will become in applying the order of operations. Try working through similar expressions, varying the numbers and operations to challenge yourself. You can also seek out practice problems in textbooks or online resources. Remember, each problem you solve is an opportunity to reinforce your understanding and refine your skills. Consistent practice is the key to mastering any mathematical concept, and evaluating expressions is no exception.

Consider these additional tips for effective practice:

  • Start with Simpler Expressions: Begin with expressions that have fewer operations and gradually increase the complexity.
  • Show Your Work: Write down each step clearly and systematically. This helps you track your progress and identify any errors.
  • Check Your Answers: Use a calculator or other tools to verify your answers. This provides immediate feedback and reinforces correct procedures.
  • Seek Help When Needed: Don't hesitate to ask for help from teachers, tutors, or classmates if you encounter difficulties.
  • Review Your Mistakes: When you make a mistake, take the time to understand why you made it and how to avoid it in the future.

By incorporating these practice strategies into your learning routine, you can develop a solid foundation in evaluating mathematical expressions and build the confidence to tackle even the most challenging problems.

Conclusion

In conclusion, evaluating the expression 7 × [(5 - 2) - 10 + (6 × 4 - 4)] has been a journey through the order of operations, highlighting the importance of PEMDAS/BODMAS in achieving accurate results. By breaking down the expression into manageable steps, simplifying parentheses, and performing operations in the correct sequence, we successfully determined the value to be 91. This process not only provides the solution but also reinforces the fundamental principles of mathematical operations. Remember, the key to mastering mathematical expressions lies in understanding the order of operations, practicing consistently, and approaching each problem with a systematic mindset. With these tools at your disposal, you can confidently navigate the world of mathematics and unlock its many possibilities. The ability to accurately evaluate expressions is a valuable asset in various fields, from science and engineering to finance and everyday problem-solving. Keep practicing, keep exploring, and keep challenging yourself to expand your mathematical horizons.