Evaluating The Expression $-4^2 imes 2 + (-4-3)^2$

Understanding the Order of Operations

In order to accurately evaluate the given mathematical expression, $-4^2 imes 2 + (-4-3)^2$, it is crucial to adhere to the order of operations, commonly remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This standardized approach ensures consistency and accuracy in mathematical calculations. First, we address any expressions within parentheses, then exponents, followed by multiplication and division (from left to right), and finally, addition and subtraction (also from left to right). By systematically applying these rules, we can break down complex expressions into manageable steps and arrive at the correct solution. The foundation of mathematical problem-solving lies in the consistent application of these rules, ensuring that every calculation is performed in the proper sequence. This not only simplifies the process but also minimizes the potential for errors, leading to a reliable and accurate final result. In the context of algebra and beyond, the order of operations serves as a fundamental principle, guiding us through complex equations and formulas with precision and confidence. The significance of PEMDAS extends beyond simple arithmetic, providing a structured approach to tackling a wide range of mathematical problems. Mastering this principle is essential for anyone pursuing studies in mathematics, science, engineering, or any field that relies on quantitative analysis. The ability to correctly interpret and apply the order of operations is a testament to one's mathematical proficiency and attention to detail. By consistently following these established guidelines, we not only arrive at correct answers but also develop a deeper understanding of the underlying mathematical principles that govern these operations. This understanding is invaluable as we progress to more advanced mathematical concepts and applications, where the order of operations remains a cornerstone of accurate and efficient problem-solving.

Step-by-Step Evaluation

To evaluate the expression $-4^2 imes 2 + (-4-3)^2$, we will meticulously follow the order of operations, also known as PEMDAS. This structured approach will ensure that we arrive at the correct answer by performing each operation in the proper sequence. First, we tackle the expression within the parentheses: $(-4-3)$. Subtracting 3 from -4 yields -7, so the expression within the parentheses simplifies to -7. Now, the original expression becomes $-4^2 imes 2 + (-7)^2$. Next, we address the exponents. It is essential to note that $-4^2$ is interpreted as $-(4^2)$, not $(-4)^2$. Therefore, we square 4 first, which gives us 16, and then apply the negative sign, resulting in -16. On the other hand, $(-7)^2$ means -7 multiplied by itself, which is (-7) × (-7) = 49. Our expression now looks like this: $-16 imes 2 + 49$. Moving on to the multiplication operation, we multiply -16 by 2, which equals -32. This transforms our expression to -32 + 49. Finally, we perform the addition. Adding -32 and 49 is equivalent to subtracting 32 from 49, which gives us 17. Therefore, the final evaluated value of the expression $-4^2 imes 2 + (-4-3)^2$ is 17. Each step in this process highlights the importance of following the order of operations to ensure accurate results. By breaking down the expression into smaller, manageable parts, we can systematically work through each operation, reducing the likelihood of errors. This step-by-step approach not only helps us arrive at the correct solution but also deepens our understanding of how different mathematical operations interact with each other. The careful execution of each step, from simplifying parentheses to performing exponents, multiplication, and finally addition, is crucial for mastering mathematical problem-solving.

Detailed Breakdown of Each Operation

Let's delve deeper into each operation within the expression $-4^2 imes 2 + (-4-3)^2$ to ensure a comprehensive understanding of the evaluation process. First, we encounter the parentheses $(-4-3)$. Inside the parentheses, we have a subtraction operation. Subtracting 3 from -4 results in -7. This step is straightforward, but its importance lies in setting the stage for subsequent operations. By simplifying the expression within the parentheses first, we adhere to the PEMDAS rule, which prioritizes operations enclosed within grouping symbols. Next, we address the exponents. Here, we have two exponential terms: $-4^2$ and $(-7)^2$. The first term, $-4^2$, requires careful attention. According to the order of operations, exponentiation precedes negation. Therefore, we square 4 first, obtaining 16, and then apply the negative sign, resulting in -16. It's crucial to differentiate this from $(-4)^2$, which would mean squaring -4, yielding 16. The second exponential term, $(-7)^2$, is more direct. Squaring -7 means multiplying -7 by itself, which equals 49. The proper handling of these exponents is essential for arriving at the correct final answer. Following the exponents, we encounter the multiplication operation: $-16 imes 2$. Multiplying -16 by 2 yields -32. This step is a simple application of multiplication rules, but it plays a critical role in combining the results of previous operations. The negative sign must be carried through accurately to ensure the correct sign in the final result. Finally, we arrive at the addition operation: -32 + 49. Adding -32 and 49 is equivalent to subtracting 32 from 49, which gives us 17. This final step combines the results of all previous operations to produce the final evaluated value of the expression. Each operation, from the initial subtraction within parentheses to the final addition, contributes to the overall solution. By understanding the nuances of each operation and following the order of operations diligently, we can confidently evaluate complex expressions and arrive at accurate results. This detailed breakdown not only clarifies the steps involved but also reinforces the importance of precision and attention to detail in mathematical calculations.

Common Mistakes to Avoid

When evaluating mathematical expressions, several common mistakes can lead to incorrect answers. Being aware of these pitfalls can significantly improve accuracy and prevent unnecessary errors. One of the most frequent mistakes involves misunderstanding the order of operations, or PEMDAS. Failing to adhere to this order can result in operations being performed in the wrong sequence, leading to a completely different outcome. For instance, in the expression $-4^2 imes 2 + (-4-3)^2$, a common error is to treat $-4^2$ as $(-4)^2$. As discussed earlier, $-4^2$ is equivalent to $-(4^2)$, which equals -16, while $(-4)^2$ equals 16. This seemingly small distinction can have a significant impact on the final result. Another common mistake occurs when dealing with negative signs. It's crucial to handle negative signs carefully, especially when squaring negative numbers. As we saw with $(-7)^2$, squaring a negative number results in a positive number. However, forgetting to apply the negative sign correctly can lead to errors in the subsequent calculations. Errors within parentheses are also prevalent. Ensuring that all operations within parentheses are performed correctly before moving on is essential. For example, in the expression $(-4-3)$, correctly subtracting 3 from -4 to get -7 is crucial. A simple arithmetic mistake here can cascade through the rest of the problem. Furthermore, mistakes can arise from neglecting to distribute negative signs properly when removing parentheses. This is particularly important when dealing with more complex expressions involving multiple sets of parentheses and negative signs. Another source of error is simply overlooking a step or miscopying numbers. It's always a good practice to double-check each step and ensure that all numbers and operations are transcribed correctly. This simple act of verification can save a lot of frustration and prevent careless mistakes. By being mindful of these common errors and taking steps to avoid them, individuals can significantly improve their accuracy in evaluating mathematical expressions. This attention to detail not only leads to correct answers but also fosters a deeper understanding of mathematical principles and problem-solving strategies.

The Solution

After a comprehensive step-by-step evaluation, the solution to the expression $-4^2 imes 2 + (-4-3)^2$ is 17. This result is obtained by meticulously following the order of operations, commonly known as PEMDAS. The process begins with simplifying the expression within the parentheses, $(-4-3)$, which yields -7. Next, we address the exponents. It's crucial to note that $-4^2$ is interpreted as $-(4^2)$, resulting in -16, while $(-7)^2$ equals 49. This distinction is vital in avoiding a common error. Following the exponents, we perform the multiplication: $-16 imes 2$, which gives us -32. Finally, we complete the evaluation by performing the addition: -32 + 49, which equals 17. This step-by-step approach not only leads to the correct answer but also underscores the importance of adhering to the established rules of mathematical operations. Each step builds upon the previous one, ensuring that the final result is accurate and reliable. The consistent application of PEMDAS ensures that the expression is simplified in a logical and consistent manner, preventing any ambiguity or misinterpretation. The solution of 17 represents the culmination of these individual operations, each performed with precision and care. By breaking down the complex expression into manageable steps, we can clearly see how each operation contributes to the final outcome. This process not only provides the correct answer but also enhances our understanding of the underlying mathematical principles. The value of 17 is not just a number; it is the result of a methodical and systematic approach to problem-solving, highlighting the power of order and precision in mathematics. This final solution serves as a testament to the importance of mastering the order of operations and applying it consistently in mathematical calculations.