Evaluating The Expression 3.4 × (-0.39 - 0.61)³ A Step-by-Step Guide

In mathematics, evaluating expressions is a fundamental skill. This article delves into the step-by-step evaluation of the expression $3.4 \times (-0.39 - 0.61)^3$. We will meticulously break down the calculation, ensuring clarity and precision in each step. This evaluation involves understanding the order of operations, handling decimal arithmetic, and working with exponents. This comprehensive approach aims to provide a clear understanding of how to solve such expressions, which is crucial for various mathematical and scientific applications.

To accurately evaluate the expression $3.4 \times (-0.39 - 0.61)^3$, we need to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order dictates the sequence in which operations should be performed to ensure the correct result. First, we tackle the operation inside the parentheses, which involves subtracting decimal numbers. Then, we deal with the exponent, cubing the result obtained from the parentheses. Finally, we complete the evaluation by multiplying the outcome by 3.4. This structured approach ensures that we address each component of the expression systematically, minimizing errors and leading to the correct final answer. By adhering to the order of operations, we can confidently navigate through the expression's complexities, making the evaluation process both efficient and accurate. Understanding and applying PEMDAS is critical not only for this particular problem but also for a wide range of mathematical computations, making it a cornerstone of algebraic proficiency. The importance of following this order cannot be overstated, as deviating from it can lead to significant discrepancies in the final result. Therefore, meticulous attention to the sequence of operations is paramount in mathematical evaluations.

Step 1: Evaluate the Parentheses

The initial step in evaluating the expression $3.4 \times (-0.39 - 0.61)^3$ is to address the operation within the parentheses. We have $-0.39 - 0.61$. This involves subtracting 0.61 from -0.39. Both numbers are negative, so we are essentially adding their absolute values and retaining the negative sign. The sum of 0.39 and 0.61 is 1.00, so the result inside the parentheses is -1.00, or simply -1. This simplification is crucial because it reduces the complexity of the expression, allowing us to proceed with the subsequent steps more easily. Correctly evaluating the parentheses is a foundational step, as any error here will propagate through the rest of the calculation, leading to an incorrect final answer. Therefore, it's essential to double-check this step to ensure accuracy. The simplicity of the result, -1, is advantageous for the next step, which involves exponentiation. This clarity helps in maintaining precision and avoiding unnecessary computational errors. By focusing on each step individually and ensuring its accuracy, we build a solid foundation for the final result. The meticulous attention to detail in this initial phase sets the stage for a successful evaluation of the entire expression.

Step 2: Handle the Exponent

Having simplified the expression inside the parentheses to -1, the next step in evaluating $3.4 \times (-0.39 - 0.61)^3$ is to address the exponent. We need to calculate $(-1)^3$, which means -1 raised to the power of 3. This is equivalent to multiplying -1 by itself three times: $(-1) \times (-1) \times (-1)$. When we multiply -1 by -1, we get 1. Then, multiplying this result by -1 again gives us -1. Therefore, $(-1)^3 = -1$. Understanding how to handle exponents, especially negative numbers raised to odd powers, is crucial in mathematics. In this case, a negative number raised to an odd power will always result in a negative number. This is because the negative signs will pair up to become positive, except for one remaining negative sign. This step is critical as it directly impacts the final result. An incorrect calculation here would lead to a completely different answer. The simplicity of the result, -1, is a direct consequence of the exponent being an odd number. This makes the subsequent multiplication step straightforward. By accurately handling the exponent, we ensure the integrity of the evaluation process and maintain the precision required for a correct final answer. This meticulous approach to each step builds confidence in the overall solution.

Step 3: Perform the Multiplication

With the parentheses and exponent evaluated, we now proceed to the final step in evaluating the expression $3.4 \times (-0.39 - 0.61)^3$, which is multiplication. We have simplified the expression to $3.4 \times (-1)$. Multiplying 3.4 by -1 is a straightforward process; it simply changes the sign of 3.4. Therefore, $3.4 \times (-1) = -3.4$. This multiplication step is crucial as it combines the results of the previous steps to yield the final answer. Ensuring accuracy in this step is paramount, as it directly reflects the correctness of the entire evaluation. The simplicity of this multiplication highlights the importance of correctly handling the preceding operations. If either the parentheses or the exponent were evaluated incorrectly, this final multiplication would lead to a wrong answer. The result, -3.4, is a decimal number, which is consistent with the problem's instruction to provide the answer as an integer or a decimal. This final step underscores the interconnectedness of mathematical operations and the need for precision at each stage. By meticulously performing the multiplication, we arrive at the final solution with confidence, knowing that we have adhered to the correct order of operations and maintained accuracy throughout the process.

After meticulously evaluating each step of the expression $3.4 \times (-0.39 - 0.61)^3$, we arrive at the final answer. We first simplified the parentheses, resulting in -1. Then, we handled the exponent, calculating $(-1)^3$ to be -1. Finally, we performed the multiplication, $3.4 \times (-1)$, which equals -3.4. Therefore, the value of the expression $3.4 \times (-0.39 - 0.61)^3$ is -3.4. This final answer is a decimal number, as requested in the problem statement. The process of evaluating this expression highlights the importance of adhering to the order of operations (PEMDAS) and maintaining precision throughout each step. Any error in the intermediate steps could lead to a different final result. The step-by-step approach ensures clarity and reduces the likelihood of mistakes. The final answer, -3.4, represents the culmination of all the calculations performed, demonstrating a comprehensive understanding of algebraic expressions and their evaluation. This result can be confidently presented as the solution to the given mathematical problem, reflecting a mastery of the fundamental principles involved. The journey from the initial expression to the final answer showcases the power of methodical problem-solving in mathematics.

In conclusion, the evaluation of the expression $3.4 \times (-0.39 - 0.61)^3$ demonstrates the significance of following the order of operations and the importance of precision in mathematical calculations. By breaking down the expression into manageable steps—first addressing the parentheses, then the exponent, and finally the multiplication—we methodically arrived at the solution, -3.4. Each step was crucial, and any error in one step would have impacted the final result. This exercise underscores the fundamental principles of algebra and the necessity of a systematic approach to problem-solving. The ability to accurately evaluate mathematical expressions is a vital skill in various fields, from science and engineering to finance and economics. The process outlined in this article provides a clear and concise method for tackling similar problems, ensuring that students and professionals alike can confidently approach mathematical evaluations. The final answer, -3.4, is not just a numerical result but a testament to the power of logical reasoning and careful execution in mathematics. This meticulous approach not only yields the correct answer but also reinforces a deeper understanding of mathematical concepts and techniques. Therefore, mastering the evaluation of expressions is a cornerstone of mathematical proficiency and a valuable asset in any analytical endeavor.