Step-by-Step Guide To Simplify The Expression $10[4 * 10 / (6^2 - 4^2) + 1]$

In this comprehensive guide, we will simplify the expression $10[4 imes 10 ext{div} (6^2 - 4^2) + 1]$ step-by-step. Mathematical expressions often appear complex at first glance, but by following the correct order of operations and breaking down the problem into smaller, manageable parts, we can arrive at the solution with confidence. This article aims to provide a clear and detailed explanation of each step involved, ensuring that even those with a basic understanding of mathematics can follow along and comprehend the process. We will cover the fundamental principles of arithmetic, including the order of operations (PEMDAS/BODMAS), and demonstrate how they are applied in this specific example. Whether you are a student looking to improve your algebra skills or someone who enjoys solving mathematical puzzles, this guide will provide valuable insights and enhance your problem-solving abilities. Let's dive into the world of mathematical expressions and uncover the elegance of simplification.

Understanding the Order of Operations (PEMDAS/BODMAS)

Before we begin simplifying the expression, it is crucial to understand the order of operations. This set of rules dictates the sequence in which mathematical operations should be performed to ensure a consistent and accurate result. The acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) and BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) are commonly used to remember this order. Both acronyms convey the same principle: calculations within parentheses or brackets are performed first, followed by exponents or orders, then multiplication and division (from left to right), and finally addition and subtraction (from left to right). Mastering the order of operations is essential for correctly simplifying any mathematical expression, as deviating from this order can lead to incorrect answers. In our specific example, we will meticulously follow PEMDAS/BODMAS to ensure the accurate simplification of the expression $10[4 imes 10 ext{div} (6^2 - 4^2) + 1]$. This involves first addressing the operations within the brackets, then exponents, followed by multiplication and division, and finally addition. By adhering to this order, we can break down the complex expression into simpler components and arrive at the correct solution systematically. Understanding and applying the order of operations is not just a procedural step; it is the foundation upon which all mathematical simplifications are built. Let's now proceed with the first step in simplifying our expression, keeping PEMDAS/BODMAS at the forefront of our minds.

Step 1: Simplify within the Parentheses

The first step in simplifying the expression $10[4 imes 10 ext{div} (6^2 - 4^2) + 1]$ according to the order of operations (PEMDAS/BODMAS) is to simplify the expression within the parentheses: $(6^2 - 4^2)$. This involves evaluating the exponents first. $6^2$ means 6 raised to the power of 2, which is $6 imes 6 = 36$. Similarly, $4^2$ means 4 raised to the power of 2, which is $4 imes 4 = 16$. Now, we substitute these values back into the parentheses: $(36 - 16)$. The next operation within the parentheses is subtraction. Subtracting 16 from 36, we get $36 - 16 = 20$. So, the expression within the parentheses simplifies to 20. This step is crucial because it reduces the complexity of the expression, making it easier to handle in subsequent steps. By addressing the parentheses first, we adhere to the fundamental principle of PEMDAS/BODMAS, ensuring that the simplification process is accurate and efficient. This methodical approach of breaking down the expression into smaller, more manageable parts is a key strategy in solving complex mathematical problems. With the parentheses now simplified to 20, we can move on to the next step, which involves addressing the operations within the brackets. This step-by-step process allows us to maintain clarity and avoid errors, ultimately leading to the correct solution. Let's proceed to the next phase of simplification and continue our journey towards the final answer.

Step 2: Simplify within the Brackets

Having simplified the parentheses, we now focus on simplifying the expression within the brackets: $[4 imes 10 ext{div} (20) + 1]$. Following the order of operations, we address multiplication and division before addition. The expression within the brackets contains both multiplication ($4 imes 10$) and division ($ extdiv} 20$). According to PEMDAS/BODMAS, multiplication and division are performed from left to right. Therefore, we first perform the multiplication $4 imes 10 = 40$. Now our expression within the brackets becomes $[40 ext{div 20 + 1]$. Next, we perform the division: $40 ext{div} 20 = 2$. This simplifies the expression within the brackets further to $[2 + 1]$. Finally, we perform the addition: $2 + 1 = 3$. So, the entire expression within the brackets simplifies to 3. This step is a critical part of the overall simplification process, as it reduces the complexity of the expression significantly. By meticulously following the order of operations within the brackets, we ensure the accuracy of our calculations and pave the way for the final step. The careful execution of multiplication, division, and addition within the brackets demonstrates the importance of a systematic approach to mathematical problem-solving. With the brackets now simplified to 3, we are just one step away from the final solution. Let's move on to the final simplification and unveil the answer.

Step 3: Final Simplification

With the expression within the brackets simplified to 3, we are now ready for the final step in simplifying the expression $10[4 imes 10 ext{div} (6^2 - 4^2) + 1]$. The expression has been reduced to $10 imes 3$. This step is straightforward and involves a single multiplication operation. Multiplying 10 by 3, we get $10 imes 3 = 30$. Therefore, the simplified form of the expression $10[4 imes 10 ext{div} (6^2 - 4^2) + 1]$ is 30. This final calculation brings our journey of simplification to a successful conclusion. By systematically applying the order of operations (PEMDAS/BODMAS) and breaking down the expression into smaller, manageable parts, we have arrived at the solution with confidence. The result, 30, is the ultimate answer to our simplification endeavor. This process highlights the importance of precision and attention to detail in mathematics. Each step, from simplifying the parentheses to the final multiplication, plays a crucial role in achieving the correct result. This exercise not only demonstrates the mechanics of simplifying expressions but also reinforces the fundamental principles of arithmetic and algebraic manipulation. Now that we have successfully simplified the expression, we can appreciate the power of mathematical tools and techniques in unraveling complex problems. The final answer, 30, stands as a testament to our meticulous approach and adherence to mathematical rules.

Conclusion

In conclusion, we have successfully simplified the expression $10[4 imes 10 ext{div} (6^2 - 4^2) + 1]$ to 30. This simplification process involved a step-by-step approach, meticulously following the order of operations (PEMDAS/BODMAS). We began by simplifying the expression within the parentheses, evaluating the exponents and performing the subtraction. Next, we moved on to simplifying the expression within the brackets, addressing multiplication and division from left to right, followed by addition. Finally, we performed the multiplication outside the brackets to arrive at the final answer. This exercise demonstrates the importance of understanding and applying the order of operations in mathematical expressions. By breaking down the problem into smaller, more manageable parts, we were able to simplify the complex expression with clarity and precision. The result, 30, is a testament to the effectiveness of a systematic approach to problem-solving in mathematics. This guide has provided a detailed explanation of each step involved, ensuring that readers can grasp the underlying principles and apply them to similar problems. Simplifying mathematical expressions is a fundamental skill that is essential for various areas of mathematics and beyond. By mastering these techniques, individuals can enhance their problem-solving abilities and gain a deeper appreciation for the elegance and logic of mathematics. We hope this comprehensive guide has been helpful in your journey to understanding and simplifying mathematical expressions. The ability to simplify expressions accurately and efficiently is a valuable asset in both academic and practical settings. Keep practicing, and you will undoubtedly become more proficient in this crucial skill.