Evaluating The Expression -0.651 - 0.969 + (0.403 - 0.697)^2

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In this article, we will dive deep into evaluating the expression -0.651 - 0.969 + (0.403 - 0.697)^2 step by step. Mathematics often involves complex expressions that require careful evaluation to arrive at the correct answer. This particular expression combines decimal subtraction and exponentiation, making it a good example to illustrate the order of operations and the importance of precision. We will break down each part of the expression, perform the necessary calculations, and provide a clear, concise final answer. This detailed explanation will help you understand not only the solution to this specific problem but also the general principles involved in evaluating similar mathematical expressions. Whether you are a student learning basic arithmetic or someone looking to refresh your math skills, this guide will provide valuable insights and a step-by-step approach to solving such problems effectively.

Understanding the Order of Operations

Before we begin, it's crucial to understand 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 we perform mathematical operations to ensure we arrive at the correct result. The order of operations is a fundamental concept in mathematics, ensuring that expressions are evaluated consistently and unambiguously. Without a standard order, the same expression could lead to different results depending on the sequence in which the operations are performed. This would create confusion and make mathematical communication unreliable. PEMDAS provides a clear guideline, starting with parentheses (or brackets), which enclose operations that must be done first. This is followed by exponents, which indicate the power to which a number is raised. Next, we perform multiplication and division from left to right, as these operations have equal precedence. Finally, we carry out addition and subtraction, also from left to right. By adhering to this order, we can systematically simplify complex expressions, breaking them down into smaller, manageable steps. This method not only helps in solving the immediate problem but also builds a strong foundation for more advanced mathematical concepts. Therefore, mastering the order of operations is essential for anyone studying mathematics, as it is a cornerstone of accurate and consistent calculations.

Applying PEMDAS to the Expression

In our expression, -0.651 - 0.969 + (0.403 - 0.697)^2, we first focus on the parentheses, then the exponent, and finally the addition and subtraction from left to right. Let's start by addressing the parentheses in our expression: (0.403 - 0.697). This step is crucial as it sets the stage for the subsequent operations. Within the parentheses, we have a subtraction operation involving two decimal numbers. To perform this subtraction accurately, we need to consider the signs and magnitudes of the numbers involved. Subtracting 0.697 from 0.403 means we are taking away a larger value from a smaller value, which will result in a negative number. The actual subtraction process involves finding the difference between the absolute values of the numbers and then applying the negative sign. This step is fundamental in simplifying the expression and allows us to move forward in the correct order. Ignoring the parentheses or performing operations in the wrong order would lead to a completely different result, highlighting the importance of following PEMDAS. Therefore, by carefully addressing the operations within the parentheses, we ensure that the expression is simplified correctly and that the final answer is accurate.

Step-by-Step Evaluation

1. Evaluate the Parentheses: (0.403 - 0.697)

To evaluate the parentheses (0.403 - 0.697), we subtract 0.697 from 0.403. This results in -0.294. The subtraction inside the parentheses is a critical first step, as it simplifies a portion of the expression and allows us to proceed with the next operation. This particular subtraction involves dealing with decimal numbers, which requires careful alignment of the decimal points to ensure accuracy. When subtracting a larger number from a smaller number, as in this case, the result is negative. The magnitude of the result is the difference between the absolute values of the two numbers. In this case, 0.697 - 0.403 equals 0.294. Since we are subtracting 0.697 from 0.403, the result is -0.294. This intermediate result is now used in the next step of the evaluation, where we will address the exponent. By focusing on the parentheses first, we adhere to the order of operations (PEMDAS) and maintain the integrity of the calculation. This step-by-step approach is essential for avoiding errors and ensuring the final answer is correct.

2. Calculate the Exponent: (-0.294)^2

Next, we calculate the exponent (-0.294)^2. This means we need to square -0.294, which is multiplying -0.294 by itself. Squaring a number means multiplying it by itself. In the case of (-0.294)^2, we are multiplying -0.294 by -0.294. When multiplying two negative numbers, the result is positive. This is a fundamental rule of arithmetic that ensures the consistency of mathematical operations. The actual multiplication of 0.294 by 0.294 involves multiplying decimal numbers, which requires careful attention to the decimal places. The product of 0.294 and 0.294 is 0.086436. This intermediate result is a positive number because we multiplied two negative numbers together. This step is crucial in simplifying the expression further and brings us closer to the final solution. By correctly applying the exponent, we transform the term (-0.294)^2 into a single numerical value, which can then be used in the subsequent addition and subtraction operations. Therefore, accurately calculating the exponent is essential for maintaining the correctness of the overall evaluation.

3. Perform the Subtraction: -0.651 - 0.969

Now, let's perform the subtraction -0.651 - 0.969. Subtracting 0.969 from -0.651 is equivalent to adding -0.969 to -0.651. This is a straightforward subtraction involving two negative decimal numbers. When subtracting a positive number from a negative number, or equivalently, adding two negative numbers, the result is always negative. The magnitude of the result is the sum of the absolute values of the two numbers. In this case, we are adding -0.651 and -0.969. The sum of their absolute values (0.651 and 0.969) is 1.62. Since both numbers are negative, the result is -1.62. This step is essential in simplifying the expression and brings us closer to the final calculation. By correctly performing the subtraction, we combine the two negative terms into a single negative term, which can then be used in the final addition operation. Therefore, accurately subtracting the numbers is crucial for maintaining the correctness of the overall evaluation and arriving at the correct final answer.

4. Final Addition: -1.62 + 0.086436

Finally, we complete the evaluation by performing the final addition: -1.62 + 0.086436. This step combines the result of the exponent calculation with the result of the subtraction. We are adding a positive decimal number (0.086436) to a negative decimal number (-1.62). This is an addition of numbers with different signs, which means we will effectively be subtracting the smaller magnitude from the larger magnitude and keeping the sign of the number with the larger magnitude. In this case, -1.62 has a larger magnitude than 0.086436, so the result will be negative. To perform the addition, we find the difference between the absolute values of the numbers. The difference between 1.62 and 0.086436 is 1.533564. Since -1.62 has a larger magnitude, the result is -1.533564. This final calculation gives us the overall value of the expression. By carefully performing the addition, we arrive at the final answer, which is a decimal number. This step concludes the evaluation process, demonstrating the importance of following the order of operations and performing each step accurately.

Detailed Calculation Breakdown

To further illustrate the calculation process, let's break down each step with detailed arithmetic:

  1. Parentheses: 0.403 - 0.697 = -0.294
  2. Exponent: (-0.294)^2 = -0.294 * -0.294 = 0.086436
  3. Subtraction: -0.651 - 0.969 = -0.651 + (-0.969) = -1.62
  4. Addition: -1.62 + 0.086436 = -1.533564

Each of these steps involves arithmetic operations with decimal numbers, which require careful alignment of the decimal points to ensure accuracy. In the first step, subtracting 0.697 from 0.403 results in -0.294. This is because we are taking away a larger value from a smaller value, resulting in a negative number. The second step involves squaring -0.294, which means multiplying it by itself. When multiplying two negative numbers, the result is positive. The product of 0.294 and 0.294 is 0.086436. In the third step, we subtract 0.969 from -0.651. This is equivalent to adding -0.969 to -0.651, which results in -1.62. Finally, we add 0.086436 to -1.62. This is an addition of numbers with different signs, so we find the difference between their absolute values and keep the sign of the number with the larger magnitude. The result is -1.533564. This detailed breakdown highlights the importance of performing each operation accurately and in the correct order to arrive at the correct final answer.

Final Answer

Therefore, the final answer to the expression -0.651 - 0.969 + (0.403 - 0.697)^2 is -1.533564. This result is obtained by systematically following the order of operations (PEMDAS) and performing each calculation with precision. The process involved several steps, including subtraction within parentheses, squaring a negative number, and adding decimal numbers with different signs. Each step was crucial in simplifying the expression and arriving at the correct solution. The negative sign in the final answer indicates that the overall value of the expression is less than zero. The decimal value represents the precise numerical result, taking into account the decimal places of the numbers involved in the calculations. This final answer is a testament to the importance of understanding and applying mathematical principles correctly. It demonstrates that by breaking down a complex expression into smaller, manageable steps, we can arrive at an accurate and reliable result. Therefore, the final answer of -1.533564 is the culmination of a thorough and meticulous evaluation process.

Conclusion

In conclusion, evaluating the expression -0.651 - 0.969 + (0.403 - 0.697)^2 requires careful application of the order of operations and precise arithmetic calculations. By following the PEMDAS rule, we first addressed the parentheses, then the exponent, followed by subtraction and addition. Each step was performed meticulously to ensure accuracy. The result of the subtraction within the parentheses was -0.294. Squaring this value gave us 0.086436. The subsequent subtraction of 0.969 from -0.651 resulted in -1.62. Finally, adding 0.086436 to -1.62 yielded the final answer of -1.533564. This detailed process demonstrates the importance of breaking down complex expressions into smaller, manageable steps. It also highlights the need for precision in each calculation to arrive at the correct final answer. Mathematics, at its core, is about solving problems systematically and logically. This example serves as a practical illustration of how mathematical principles can be applied to evaluate complex expressions. By understanding the order of operations and practicing arithmetic skills, one can confidently tackle similar problems and achieve accurate results. Therefore, this evaluation serves as a valuable exercise in mathematical problem-solving and reinforces the fundamental concepts of arithmetic and algebra.