Evaluate -7² - 28 ÷ 4 - 2 By Hand A Step-by-Step Guide

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In the realm of mathematics, mastering the order of operations is crucial for accurate calculations. This article provides a detailed, step-by-step guide on how to evaluate the expression -7² - 28 ÷ 4 - 2 by hand. We'll break down each step, explaining the reasoning behind it, and ensuring a clear understanding of the process. By the end of this guide, you'll be equipped to tackle similar mathematical expressions with confidence and precision. Whether you're a student learning the fundamentals or someone looking to refresh your math skills, this article will serve as a valuable resource. Let's dive in and demystify the process of manual evaluation.

Understanding the Order of Operations (PEMDAS/BODMAS)

Before we begin evaluating the expression, it's essential to understand the order of operations, often remembered by the acronyms PEMDAS or BODMAS. These acronyms dictate the sequence in which mathematical operations should be performed:

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

This hierarchy ensures that mathematical expressions are evaluated consistently, regardless of who is solving them. Without a standardized order, the same expression could yield different results, leading to confusion and errors. By adhering to PEMDAS/BODMAS, we maintain clarity and accuracy in mathematical calculations. In our expression, -7² - 28 ÷ 4 - 2, we'll follow this order meticulously to arrive at the correct answer. Understanding this foundational principle is key to success in mathematics and forms the backbone of our evaluation process.

Step 1: Evaluating the Exponent (-7²)

The first step in evaluating the expression -7² - 28 ÷ 4 - 2 is to address the exponent. It's crucial to understand that -7² is interpreted as -(7²), not (-7)². This distinction is paramount to achieving the correct answer. Let's break it down:

  • 7² means 7 multiplied by itself, which is 7 * 7 = 49.
  • Therefore, -7² is the negative of 49, resulting in -49.

This step highlights the importance of careful interpretation and attention to detail in mathematical expressions. A seemingly small difference in notation can lead to a significant difference in the final result. Many common errors in math arise from misinterpreting the order of operations or the nuances of notation. By correctly evaluating the exponent as -49, we set the stage for the subsequent steps in our calculation. This understanding of exponential notation and its interaction with negative signs is a fundamental building block for more advanced mathematical concepts.

Step 2: Performing the Division (28 ÷ 4)

Following the order of operations, after handling the exponent, we move on to division. In the expression -7² - 28 ÷ 4 - 2, the division operation is 28 ÷ 4. This is a straightforward calculation:

  • 28 divided by 4 equals 7.

Therefore, 28 ÷ 4 = 7. This step is relatively simple but crucial in the overall evaluation process. It demonstrates how each operation contributes to the final answer. By performing the division correctly, we reduce the complexity of the expression and bring ourselves closer to the final solution. This step reinforces the importance of mastering basic arithmetic operations as a foundation for more complex mathematical problems. Accuracy in division, like in all operations, is paramount to achieving a correct final result. With the division completed, we can now proceed to the final steps involving subtraction.

Step 3: Performing Subtractions from Left to Right (-49 - 7 - 2)

With the exponent and division operations completed, we are now left with a series of subtractions: -49 - 7 - 2. According to the order of operations, addition and subtraction are performed from left to right. This means we first subtract 7 from -49, and then subtract 2 from the result.

  • -49 - 7 = -56
  • -56 - 2 = -58

Therefore, the result of the subtractions is -58. This step highlights the importance of following the correct order when dealing with multiple additions and subtractions. Performing them in the wrong order would lead to an incorrect answer. The left-to-right rule ensures consistency and accuracy in our calculations. This step also reinforces the concept of negative numbers and how they interact with subtraction. Understanding these principles is vital for success in algebra and beyond. With all operations now performed, we have arrived at the final solution for the expression.

Final Result: -58

By meticulously following the order of operations (PEMDAS/BODMAS) and performing each step with accuracy, we have successfully evaluated the expression -7² - 28 ÷ 4 - 2. The final result is -58.

This exercise demonstrates the importance of understanding and applying the correct order of operations in mathematics. Each step, from evaluating the exponent to performing the division and subtractions, played a crucial role in arriving at the correct answer. A single error in any step could have led to a different, incorrect result. This underscores the need for careful attention to detail and a thorough understanding of mathematical principles.

Furthermore, this example illustrates how seemingly complex expressions can be broken down into simpler, manageable steps. By tackling each operation individually and in the correct sequence, we can solve even intricate problems with confidence. This step-by-step approach is a valuable strategy for problem-solving in mathematics and other disciplines.

In conclusion, the evaluation of -7² - 28 ÷ 4 - 2 to -58 showcases the power of methodical calculation and the importance of adhering to fundamental mathematical rules. This skill is not only essential for academic success but also for numerous real-world applications where accurate calculations are paramount.