Solving Complex Math Problems Step-by-Step Guide For (65371 - (632 × 13 + 256 - 208)) × 213
Introduction
In this article, we will delve into the step-by-step solution of the mathematical expression (65,371 - (632 × 13 + 256 - 208)) × 213. This problem involves a combination of arithmetic operations, including multiplication, addition, and subtraction, nested within parentheses. 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 accurately solving this equation. We will break down each step in detail, ensuring clarity and comprehension for readers of all backgrounds. This comprehensive guide aims to not only provide the solution but also to enhance your problem-solving skills in mathematics. By following along with this explanation, you will gain confidence in tackling complex arithmetic problems and appreciate the logical progression required to arrive at the correct answer. Let's embark on this mathematical journey together and unlock the solution to this intriguing problem.
Step-by-Step Breakdown
1. Solve the Innermost Parentheses First: 632 × 13
In the journey of solving complex mathematical expressions, the first critical step is to address the innermost parentheses. Within our expression, (65,371 - (632 × 13 + 256 - 208)) × 213, the operation 632 × 13 takes precedence. Multiplication forms the core of this initial calculation. To perform this multiplication effectively, we break it down into manageable parts, ensuring accuracy and clarity. This multiplication is not just a standalone calculation; it’s the foundation upon which the rest of the solution is built. The result of this step will significantly influence the subsequent operations, making it imperative to handle it with precision. Through careful execution of this multiplication, we set the stage for the remainder of the problem, ensuring a logical and methodical approach towards the final answer. Understanding and mastering this initial step is crucial, as it embodies the fundamental principles of order of operations in mathematics.
When multiplying 632 by 13, we can use the standard multiplication method:
632
× 13
-----
1896 (632 × 3)
6320 (632 × 10)
-----
8216
Thus, 632 × 13 = 8216. This is the first key result we need.
2. Continue within the Parentheses: 8216 + 256
Following the successful completion of the initial multiplication within the parentheses, we now proceed to the next operation in our mathematical journey. Our expression now looks like (65,371 - (8216 + 256 - 208)) × 213. The subsequent step involves addition, specifically, 8216 + 256. Addition, a fundamental arithmetic operation, is crucial in piecing together the various components of our equation. By performing this addition, we continue to simplify the contents within the parentheses, gradually moving closer to a consolidated value. This step is not just about adding numbers; it's about maintaining the flow of the problem-solving process, ensuring that each operation logically leads to the next. Accuracy in addition is paramount here, as any error can propagate through the remaining calculations, leading to an incorrect final answer. Therefore, we approach this step with diligence, setting the stage for further simplification and eventual resolution of the entire expression. The addition process is a critical link in the chain of operations that will ultimately unravel the solution.
Now, we add 8216 and 256:
8216
+ 256
-----
8472
So, 8216 + 256 = 8472. This intermediate result is essential for the next step.
3. Complete the Parenthetical Operations: 8472 - 208
Having successfully executed the addition within the parentheses, our mathematical journey progresses to the next critical operation: subtraction. The expression now stands as (65,371 - (8472 - 208)) × 213. The focus shifts to subtracting 208 from 8472. Subtraction, a fundamental arithmetic operation, plays a crucial role in refining the value within the parentheses. This step is not merely about reducing a number; it's about carefully and precisely determining the difference, which will significantly impact the subsequent calculations. The accuracy of this subtraction is paramount, as any error could compromise the final result. As we diligently perform this operation, we move closer to isolating a single value within the parentheses, thereby simplifying the overall expression. This subtraction is a key component in our step-by-step approach, allowing us to methodically unravel the complexities of the equation. By mastering this subtraction, we reinforce our understanding of arithmetic principles and prepare ourselves for the final stages of problem-solving.
Next, we subtract 208 from 8472:
8472
- 208
-----
8264
Therefore, 8472 - 208 = 8264. With this subtraction, we have simplified the expression inside the parentheses to a single value.
4. Subtract from the First Number: 65,371 - 8264
Having diligently simplified the expression within the parentheses, we now transition to the next crucial step in our mathematical journey. The expression at this stage is (65,371 - 8264) × 213. The operation that commands our attention is the subtraction of 8264 from 65,371. Subtraction, a fundamental arithmetic process, is paramount in determining the difference between these two significant numbers. This step is not just about deducting one value from another; it’s about precisely calculating the remainder, which will serve as a critical component in the final calculation. Accuracy in this subtraction is of utmost importance, as any error will directly impact the final answer. As we meticulously perform this operation, we bring ourselves closer to the ultimate solution, reinforcing our understanding of arithmetic principles. This subtraction acts as a bridge, connecting the simplified parenthetical value to the external multiplier, setting the stage for the concluding multiplication. By mastering this subtraction, we demonstrate proficiency in handling large numbers and complex expressions.
Now, we subtract 8264 from 65,371:
65371
- 8264
-----
57107
Thus, 65,371 - 8264 = 57,107. This result simplifies the expression to a straightforward multiplication.
5. Final Multiplication: 57,107 × 213
Having successfully navigated through the parentheses and performed the subtraction, we now arrive at the concluding step of our mathematical journey: multiplication. The expression has been simplified to 57,107 × 213. This final multiplication is the culmination of all the preceding operations, and it will yield the ultimate solution to our problem. Multiplication, a fundamental arithmetic process, serves here as the grand finale, combining two significant numbers to produce the final answer. The accuracy of this multiplication is paramount, as it represents the culmination of all our efforts. This step is not just about multiplying numbers; it's about bringing closure to the entire equation, showcasing our mastery of arithmetic principles. As we meticulously perform this multiplication, we complete the problem-solving process, reinforcing our understanding of mathematical operations and their logical progression. By conquering this final step, we not only arrive at the solution but also demonstrate our proficiency in handling complex calculations with confidence and precision.
To find the final answer, we multiply 57,107 by 213:
57107
× 213
--------
171321 (57107 × 3)
571070 (57107 × 10)
11421400 (57107 × 200)
--------
12163791
Therefore, 57,107 × 213 = 12,163,791.
Conclusion
In conclusion, the step-by-step solution of the mathematical expression (65,371 - (632 × 13 + 256 - 208)) × 213 has led us to the final answer of 12,163,791. This journey through the intricacies of arithmetic operations has highlighted the importance of following the order of operations (PEMDAS/BODMAS) to ensure accuracy. By meticulously breaking down the problem into manageable steps, we have demonstrated how to tackle complex equations with confidence and precision. Each operation, from multiplication and addition to subtraction, played a crucial role in simplifying the expression and ultimately arriving at the correct solution. This exercise not only reinforces our understanding of mathematical principles but also enhances our problem-solving skills. The ability to methodically approach and solve complex calculations is a valuable asset in various fields, and the knowledge gained from this detailed solution will undoubtedly prove beneficial in future mathematical endeavors. We trust that this comprehensive guide has provided clarity and insight into the world of arithmetic operations, empowering you to tackle similar challenges with ease and proficiency.
