Step-by-Step Evaluation Of The Expression 48 ÷ 6 + (5 × 6) × 13 + 6
In mathematics, evaluating expressions requires a systematic approach to ensure accuracy. Following the order of operations, often remembered by the acronym PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction), is crucial for obtaining the correct result. In this article, we will meticulously evaluate the expression 48 ÷ 6 + (5 × 6) × 13 + 6, providing a step-by-step breakdown and verification of each stage. This detailed examination not only helps in understanding the solution but also reinforces the importance of adhering to mathematical principles. Our goal is to provide a clear, concise, and comprehensive explanation that caters to both students and anyone interested in enhancing their mathematical skills. So, let's delve into the intricacies of this expression and unravel its solution.
Step 1: Original Expression
In this initial step, we begin with the expression as it is originally presented: 48 ÷ 6 + (5 × 6) × 13 + 6. This serves as the foundation for our step-by-step evaluation. It's crucial to start by clearly stating the original expression to ensure there's no ambiguity about what we're trying to solve. This step may seem straightforward, but it's an essential part of the process as it sets the stage for the subsequent operations. By explicitly writing out the expression, we make it easier to follow the logical progression of steps and to verify the correctness of each calculation. Moreover, this clear starting point helps prevent errors that can arise from misinterpreting the original problem. Therefore, documenting the original expression is not just a formality; it's a fundamental aspect of sound mathematical practice, ensuring transparency and accuracy throughout the evaluation process. This attention to detail is particularly important in complex expressions where multiple operations are involved, as it provides a reference point for tracking progress and identifying any potential mistakes along the way. By emphasizing the importance of this initial step, we highlight the methodical approach necessary for successful mathematical problem-solving.
Step 2: Simplify Parentheses and Division
The second step involves simplifying the expression by addressing the parentheses and division operations first. Following the order of operations (PEMDAS/BODMAS), parentheses take precedence. Thus, we calculate (5 × 6), which equals 30. Simultaneously, we perform the division operation 48 ÷ 6, resulting in 8. Substituting these values back into the expression, we get 8 + 30 × 13 + 6. This step is pivotal as it reduces the complexity of the expression, making it easier to manage in the subsequent calculations. By handling the parentheses and division early on, we're essentially breaking down the problem into more manageable parts. This approach not only simplifies the arithmetic but also minimizes the chances of error. Each operation is performed independently, and the results are then integrated back into the overall expression, maintaining a clear and logical flow. This methodical reduction of complexity is a key strategy in mathematical problem-solving, particularly when dealing with expressions that involve multiple operations. Furthermore, this step demonstrates the practical application of the order of operations, a fundamental principle in mathematics that ensures consistent and accurate results.
Step 3: Perform Multiplication
Step 3 focuses on performing the multiplication operation. Following the order of operations, multiplication comes before addition and subtraction. In the expression 8 + 30 × 13 + 6, we identify the multiplication as 30 × 13. Calculating this product yields 390. Substituting this result back into the expression, we now have 8 + 390 + 6. This step is crucial because it further simplifies the expression by reducing the number of operations required. By addressing the multiplication before moving on to addition, we adhere strictly to the mathematical conventions that govern the order in which operations should be performed. This not only ensures the correctness of the solution but also reflects a disciplined approach to problem-solving. The multiplication step transforms the expression from one involving multiplication and addition to one that solely involves addition, making the final calculations more straightforward. This incremental simplification is a hallmark of effective mathematical technique, allowing us to tackle complex problems by breaking them down into smaller, more manageable steps. The precision in performing this multiplication is essential, as any error here would propagate through the remaining steps, leading to an incorrect final answer.
Step 4: Add the First Two Numbers
In the fourth step, we proceed with the addition operations. We have the expression 8 + 390 + 6. To simplify, we first add the first two numbers: 8 + 390, which equals 398. The expression now becomes 398 + 6. This step is a straightforward application of addition, but it's important to execute it carefully to maintain accuracy. By adding the numbers sequentially, we avoid the potential for errors that can arise from trying to perform multiple additions simultaneously. This methodical approach is a key aspect of problem-solving in mathematics, ensuring that each operation is carried out correctly before moving on to the next. The result of this addition, 398, represents a significant simplification of the original expression, bringing us closer to the final answer. This incremental progress highlights the effectiveness of breaking down complex calculations into smaller, more manageable steps. The focus here is on precision and accuracy, as the intermediate result will directly influence the final outcome.
Step 5: Final Addition
The final step in evaluating the expression 48 ÷ 6 + (5 × 6) × 13 + 6 is the last addition. We have the simplified expression 398 + 6. Performing this addition yields the final result: 404. This step concludes the sequence of operations, providing the solution to the original mathematical expression. The simplicity of this final addition belies the importance of the preceding steps, each of which played a crucial role in simplifying the expression and arriving at the correct answer. This final calculation serves as a verification of all the previous steps, demonstrating the accuracy of our methodical approach. The result, 404, represents the culmination of our efforts, and its correctness underscores the importance of adhering to the order of operations and maintaining precision throughout the evaluation process. This final step not only provides the numerical answer but also reinforces the value of a systematic approach to mathematical problem-solving.
Conclusion
In conclusion, the step-by-step evaluation of the expression 48 ÷ 6 + (5 × 6) × 13 + 6 has demonstrated the importance of following the order of operations (PEMDAS/BODMAS) to arrive at the correct result. By meticulously breaking down the expression into manageable steps, we've shown how to simplify complex calculations and minimize the risk of errors. Starting with the original expression, we first addressed the parentheses and division, followed by multiplication, and finally, the addition operations. Each step was crucial in transforming the expression until we reached the final answer of 404. This process not only highlights the mechanics of solving mathematical expressions but also reinforces the significance of a systematic and disciplined approach. The detailed explanation provided for each step aims to enhance understanding and build confidence in mathematical problem-solving skills. Whether you're a student learning the fundamentals or someone looking to brush up on your math, the principles demonstrated here are invaluable for tackling a wide range of mathematical challenges. By mastering the order of operations and practicing a methodical approach, you can confidently evaluate expressions and achieve accurate results.
