Evaluating -(2-2^2)^2-3*5 A Step-by-Step Guide

Understanding the Expression

To accurately evaluate the given expression, $-\left(2-2^2\right)^2-3 \cdot 5$, it is imperative to meticulously follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This systematic approach ensures that we arrive at the correct result by performing operations in the appropriate sequence. The expression involves several mathematical operations, including exponentiation, subtraction, multiplication, and the application of a negative sign. Each of these operations must be handled with precision to avoid errors. Before diving into the step-by-step evaluation, let's break down the expression into its components to clearly understand what needs to be computed. The expression consists of two main parts separated by a subtraction: $-\left(2-2^2\right)^2$ and $3 \cdot 5$. The first part, $-\left(2-2^2\right)^2$, contains a set of parentheses with operations inside, an exponent outside the parentheses, and a negative sign applied to the entire term. The second part, $3 \cdot 5$, is a simple multiplication. We will address the operations inside the parentheses first, then the exponent, followed by any multiplications or divisions, and finally, additions or subtractions. By adhering to this structured approach, we can methodically simplify the expression and determine its final value with confidence. The parentheses in the expression $-\left(2-2^2\right)^2-3 \cdot 5$ indicate that we must first evaluate the contents within them. This is a critical first step in the order of operations. Inside the parentheses, we have $2-2^2$. Here, we encounter another layer of operations, necessitating that we address the exponent before the subtraction. The term $2^2$ represents 2 raised to the power of 2, which means 2 multiplied by itself. So, $2^2$ equals 4. Now, we can substitute this value back into the parentheses, giving us $2 - 4$. Performing this subtraction, we find that $2 - 4 = -2$. Therefore, the expression inside the parentheses simplifies to $-2$. This intermediate result is crucial as we move forward in the evaluation process. We have successfully handled the exponent within the parentheses and completed the subtraction, which prepares us for the next step: addressing the exponent outside the parentheses. By following this methodical approach, we minimize the risk of errors and ensure the accuracy of our final result. Understanding the structure of the expression and systematically addressing each component is key to successful evaluation.

Step-by-Step Evaluation

Now that we've simplified the expression inside the parentheses, the next step in evaluating $-\left(2-2^2\right)^2-3 \cdot 5$ is to address the exponent outside the parentheses. Recall that we found the expression inside the parentheses to be $-2$. Thus, we now have $-\left(-2\right)^2$. The exponent 2 indicates that we need to square the value inside the parentheses, which means multiplying $-2$ by itself. Mathematically, this is expressed as $(-2)^2 = (-2) \cdot (-2)$. When we multiply two negative numbers, the result is a positive number. Therefore, $(-2) \cdot (-2) = 4$. Substituting this result back into our expression, we now have $-4$. It is crucial to observe the negative sign in front of the parentheses, which we have carried down. This negative sign applies to the entire result of the squared term. Thus, $-\left(-2\right)^2$ simplifies to $-4$. This step is pivotal as it correctly handles the exponent and the negative sign, which are common areas for errors if not addressed carefully. With this simplification, our expression now looks like $-4 - 3 \cdot 5$. We have successfully dealt with the parentheses and the exponent, and we are ready to move on to the next operation in the order of operations. The next operation to consider in evaluating the expression $-4 - 3 \cdot 5$ is the multiplication. According to the order of operations (PEMDAS), multiplication and division should be performed before addition and subtraction. In this case, we have $3 \cdot 5$, which represents 3 multiplied by 5. Performing this multiplication, we find that $3 \cdot 5 = 15$. Now, we substitute this result back into our expression, which becomes $-4 - 15$. This simplification is crucial because it reduces the expression to a simple subtraction problem. By correctly performing the multiplication first, we ensure that we are following the order of operations accurately. The expression $-4 - 15$ is now much easier to evaluate, and we are one step closer to finding the final answer. The importance of adhering to the order of operations cannot be overstated, as it guarantees that we arrive at the correct result. With the multiplication completed, we are ready to perform the final subtraction to find the value of the expression.

Final Calculation

Having simplified the expression to $-4 - 15$, the final step in evaluating $-\left(2-2^2\right)^2-3 \cdot 5$ is to perform the subtraction. The expression $-4 - 15$ can be interpreted as starting at $-4$ on the number line and moving 15 units further in the negative direction. This is equivalent to adding $-15$ to $-4$. Therefore, we have $-4 + (-15)$. When adding two negative numbers, we add their absolute values and keep the negative sign. The absolute value of $-4$ is 4, and the absolute value of $-15$ is 15. Adding these gives us $4 + 15 = 19$. Since both numbers are negative, the result is also negative, so $-4 - 15 = -19$. Thus, the final value of the expression is $-19$. This completes the evaluation of the given expression. We have followed the order of operations meticulously, simplifying each part of the expression step-by-step to arrive at the correct result. This methodical approach not only ensures accuracy but also provides a clear and understandable solution process. Understanding how to evaluate mathematical expressions is a fundamental skill in mathematics. By breaking down complex expressions into simpler steps and adhering to the order of operations, we can confidently solve a wide range of mathematical problems. The key to success lies in careful attention to detail and a systematic approach. In summary, the expression $-\left(2-2^2\right)^2-3 \cdot 5$ simplifies to $-19$ through the application of the order of operations and careful arithmetic. This result underscores the importance of understanding and applying mathematical principles correctly. The process of evaluating expressions is not just about finding the correct answer; it's also about developing logical thinking and problem-solving skills that are applicable in various areas of mathematics and beyond.

Conclusion

In conclusion, evaluating the expression $-\left(2-2^2\right)^2-3 \cdot 5$ requires a thorough understanding and application of the order of operations. By following the PEMDAS rule, which prioritizes Parentheses, Exponents, Multiplication and Division, and finally Addition and Subtraction, we methodically simplified the expression to arrive at the final answer. The initial step involved addressing the operations within the parentheses, where we first computed the exponent $2^2$ and then performed the subtraction. This was followed by handling the exponent outside the parentheses, which involved squaring the result from the parentheses. Next, we performed the multiplication operation. Finally, we completed the evaluation by performing the subtraction, which yielded the result of $-19$. Each step in this process is crucial, and any deviation from the order of operations can lead to an incorrect result. The exercise highlights the importance of precision and attention to detail in mathematical calculations. By breaking down a complex expression into smaller, manageable steps, we can minimize the risk of errors and ensure accuracy. Moreover, this methodical approach not only helps in solving the problem at hand but also enhances our understanding of mathematical principles and problem-solving strategies. The process of evaluation is a fundamental skill in mathematics and forms the basis for more advanced mathematical concepts. Mastery of this skill enables us to confidently tackle a wide range of mathematical problems and apply these principles in various fields. Ultimately, the successful evaluation of the expression $-\left(2-2^2\right)^2-3 \cdot 5$ to $-19$ serves as a testament to the power of methodical thinking and the importance of adhering to mathematical rules and conventions. This skill is not only crucial for academic success but also for practical applications in everyday life, where problem-solving and analytical thinking are highly valued. Therefore, understanding and mastering the order of operations is an invaluable asset in both mathematics and broader contexts.