Step-by-Step Guide Evaluate $[(16+24) \div 8-3] \times 7$

In the realm of mathematics, the ability to evaluate expressions accurately is a foundational skill. Often, these expressions involve multiple operations and parentheses, requiring a systematic approach to ensure the correct solution. In this article, we will delve into the process of evaluating a specific mathematical expression: $[(16+24) \div 8-3] \times 7$. We will break down each step, adhering to the order of operations, and provide a clear, concise explanation for each calculation. Whether you're a student learning the basics or someone looking to refresh your mathematical skills, this guide will offer valuable insights into expression evaluation.

Understanding the Order of Operations

To evaluate mathematical expressions correctly, we must adhere to the order of operations, often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This order dictates the sequence in which operations should be performed to arrive at the accurate answer. Ignoring this order can lead to significantly different and incorrect results. In our given expression, $[(16+24) \div 8-3] \times 7$, we have parentheses, division, subtraction, and multiplication. According to PEMDAS, we must first address the operations within the parentheses, followed by division, subtraction, and finally, multiplication. This structured approach ensures that we handle each operation in its proper sequence, preventing confusion and errors. By consistently applying the order of operations, we can confidently tackle even the most complex expressions and achieve the correct solution. Remember, PEMDAS isn't just a rule; it's the foundation of mathematical consistency and accuracy.

Step 1: Simplifying the Innermost Parentheses

Our initial focus in evaluating this expression is to simplify the innermost parentheses: $(16+24)$. Within this set of parentheses, we have a simple addition operation. Adding 16 and 24 gives us 40. This step is crucial as it reduces the complexity of the expression, allowing us to proceed with the remaining operations more smoothly. By resolving the parentheses first, we adhere to the order of operations (PEMDAS), which prioritizes operations enclosed within parentheses. This principle is not just a rule but a fundamental aspect of mathematical syntax, ensuring that expressions are interpreted and solved consistently. The result, 40, now replaces the original expression within the parentheses, simplifying the overall equation and paving the way for the subsequent steps in our evaluation process. This methodical approach, starting with the innermost parentheses, is key to accurately solving complex mathematical expressions.

Step 2: Performing Division

After simplifying the parentheses, our next step in evaluating the expression $[(16+24) \div 8-3] \times 7$ is to perform the division operation. We now have the expression $[40 \div 8 - 3] \times 7$. Focusing on the brackets, we encounter the division $40 \div 8$. Dividing 40 by 8 yields 5. This division operation is a critical step in isolating the value within the brackets before moving on to the next operation. Adhering to the order of operations (PEMDAS), division and multiplication take precedence over addition and subtraction. Therefore, resolving the division at this stage is essential for maintaining the integrity of the equation and ensuring an accurate final answer. The result of this division, 5, simplifies the expression further, bringing us closer to the final solution. This methodical progression through the order of operations is what allows us to tackle complex calculations with confidence and precision.

Step 3: Subtraction within the Brackets

Having completed the division, the next operation to tackle in evaluating the expression $[(16+24) \div 8-3] \times 7$ is the subtraction within the brackets. With the division resolved, our expression now reads $[5 - 3] \times 7$. Performing the subtraction, we subtract 3 from 5, resulting in 2. This step is vital as it further simplifies the contents within the brackets, bringing us closer to a single value that can then be used for the final multiplication. According to the order of operations (PEMDAS), subtraction and addition are performed after parentheses, exponents, multiplication, and division. By addressing the subtraction at this stage, we continue to follow the established mathematical conventions, ensuring that our solution is accurate and consistent with standard mathematical practices. The result of this subtraction, 2, represents the simplified value of the expression within the brackets, setting the stage for the final calculation.

Step 4: Final Multiplication

With the expression within the brackets now simplified to 2, the final step in evaluating the expression $[(16+24) \div 8-3] \times 7$ is to perform the multiplication. We have the simple operation $2 \times 7$. Multiplying 2 by 7 gives us 14. This multiplication is the culmination of all the previous steps, where we systematically simplified the expression according to the order of operations (PEMDAS). By adhering to this order, we have ensured that each operation was performed in its proper sequence, leading us to the correct final answer. The result, 14, is the solution to the original expression. This final step underscores the importance of following mathematical conventions and highlights how a methodical approach can lead to accurate results, even in seemingly complex calculations. Therefore, the value of the expression $[(16+24) \div 8-3] \times 7$ is 14.

Conclusion

In conclusion, the process of evaluating mathematical expressions, such as $[(16+24) \div 8-3] \times 7$, requires a systematic approach rooted in the order of operations (PEMDAS). By methodically simplifying each part of the expression—starting with the innermost parentheses, proceeding with division, then subtraction, and finally multiplication—we arrive at the correct solution. This step-by-step approach not only ensures accuracy but also provides a clear understanding of the mathematical principles at play. The final answer, 14, is a testament to the power of PEMDAS and the importance of following mathematical conventions. Whether you're a student tackling algebraic problems or simply seeking to enhance your mathematical skills, mastering the order of operations is crucial for success. Remember, mathematics is not just about numbers; it's about understanding the logical sequence and relationships that govern these numbers. By consistently applying these principles, we can confidently navigate the complexities of mathematical expressions and achieve accurate results.