Evaluating The Expression -0.651 - 0.969 + (0.403 - 0.697)^2
Introduction
In this article, we will delve into the step-by-step evaluation of the mathematical expression: -0.651 - 0.969 + (0.403 - 0.697)^2. This exercise involves a combination of arithmetic operations, including subtraction and exponentiation. Our goal is to arrive at the final answer as an integer or a decimal, ensuring precision without any rounding. Understanding the order of operations, often remembered by the acronym PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction), is crucial for accurate evaluation. Let's embark on this mathematical journey, breaking down each step to ensure clarity and precision. This article aims to not only solve the given expression but also to reinforce the fundamental principles of arithmetic operations. By meticulously working through each step, we will gain a deeper understanding of how these operations interact and influence the final result. The ability to accurately evaluate such expressions is a cornerstone of mathematical proficiency, applicable in various fields ranging from basic arithmetic to advanced calculus. Our detailed approach will serve as a guide, making it easier for readers to follow along and apply these principles to similar problems. Through careful calculation and a systematic approach, we will unravel the intricacies of this expression and arrive at the correct solution. Let's begin by addressing the operations within the parentheses, setting the stage for the subsequent steps in our evaluation.
Step 1: Simplify within the Parentheses
The first step in evaluating the expression is to simplify the terms within the parentheses: (0.403 - 0.697). This involves a simple subtraction. Subtracting 0.697 from 0.403 gives us: 0.403 - 0.697 = -0.294. This result is crucial as it sets the stage for the next operation, which is squaring this value. Parentheses play a vital role in mathematical expressions, dictating the order in which operations are performed. By addressing the parentheses first, we ensure that the subsequent steps are based on the correct intermediate result. The negative sign in our result, -0.294, is significant and must be carefully considered when squaring the number. A common mistake is to overlook the negative sign, which can lead to an incorrect final answer. Now that we have simplified the expression within the parentheses, we are ready to move on to the next operation: squaring the result. This step will involve multiplying -0.294 by itself, which will give us a positive value. Understanding the properties of negative numbers and their behavior when squared is essential for accurate mathematical calculations. By performing this initial subtraction carefully, we have laid a solid foundation for the rest of the evaluation. This step-by-step approach not only helps in solving the problem correctly but also reinforces the importance of following the order of operations in mathematics. Let's proceed to the next step, where we will square the result obtained from the parentheses.
Step 2: Exponentiation
Having simplified the expression within the parentheses, our next step is to address the exponent. We need to calculate (-0.294)^2, which means multiplying -0.294 by itself. When a negative number is squared, the result is always positive because the product of two negative numbers is positive. So, we calculate: -0.294 * -0.294 = 0.086436. This exponentiation step is critical in determining the final value of the expression. Exponents indicate the power to which a number is raised, and in this case, squaring -0.294 means multiplying it by itself. The result, 0.086436, is a positive decimal, and it represents the contribution of the squared term to the overall expression. Understanding how exponents work is fundamental in mathematics, and it is particularly important when dealing with negative numbers and decimals. A common mistake is to incorrectly apply the exponent, either by forgetting the negative sign or by miscalculating the multiplication. By carefully squaring -0.294, we have ensured that this part of the expression is correctly evaluated. This result will now be used in the subsequent addition and subtraction operations to arrive at the final answer. Exponentiation often comes before addition and subtraction in the order of operations, making it essential to address this step before moving on to the remaining arithmetic operations. With the exponentiation complete, we can now proceed to the final steps of adding and subtracting the remaining terms in the expression. Let's move on to the next step, where we will incorporate this result into the overall expression.
Step 3: Perform Subtraction and Addition
Now that we have calculated the squared term, 0.086436, we can substitute it back into the original expression. The expression now looks like this: -0.651 - 0.969 + 0.086436. The remaining operations are subtraction and addition, which are performed from left to right. First, we subtract 0.969 from -0.651: -0.651 - 0.969 = -1.62. Next, we add 0.086436 to -1.62: -1.62 + 0.086436 = -1.533564. This step combines the results of the previous operations to arrive at the final value. Addition and subtraction are inverse operations, and their order matters when performed sequentially from left to right. By carefully performing these operations, we ensure that the final result accurately reflects the combined effect of all the terms in the expression. A common mistake is to perform addition before subtraction, which can lead to an incorrect answer. Following the correct order of operations is crucial for mathematical accuracy. The negative sign in the final result indicates that the overall value of the expression is negative, which is a consequence of the negative terms dominating the positive term. This final value represents the complete evaluation of the given expression, taking into account all the operations and their respective results. By systematically working through each step, we have arrived at a precise answer without any rounding. Let's now state the final answer, summarizing our step-by-step evaluation. With the subtraction and addition completed, we have successfully evaluated the expression.
Final Answer
After performing all the necessary calculations, we find that the value of the expression -0.651 - 0.969 + (0.403 - 0.697)^2 is -1.533564. This result is obtained by meticulously following the order of operations, ensuring accuracy at each step. The final answer is a decimal number, which is consistent with the nature of the original terms in the expression. By breaking down the problem into smaller, manageable steps, we were able to avoid errors and arrive at the correct solution. The process involved simplifying within parentheses, exponentiation, and then performing subtraction and addition from left to right. Each step played a crucial role in determining the final value, and by carefully executing each operation, we have demonstrated a comprehensive understanding of arithmetic principles. This final answer represents the culmination of our efforts, and it serves as a testament to the importance of precision and attention to detail in mathematical calculations. The negative sign indicates that the overall value of the expression is negative, which is a reflection of the relative magnitudes of the negative and positive terms. This comprehensive evaluation not only provides the answer but also reinforces the fundamental concepts of arithmetic operations and their application in solving mathematical expressions. With the final answer determined, we have successfully completed the evaluation of the given expression. The result, -1.533564, is the precise value of the expression, achieved through a systematic and careful approach.
Therefore, the final answer is: -1.533564
