Calculating The Complex Mathematical Expression

In this article, we will delve into the step-by-step calculation of the complex mathematical expression: $\frac{\frac{11,000(\frac{0.09}{12})}{[1-(1+\frac{0.09}{12})^{-12 \times 6}]}}{198.28}$. This expression involves several arithmetic operations, including division, multiplication, exponentiation, and subtraction. We will break down the expression into smaller, manageable parts, calculate each part individually, and then combine the results to obtain the final answer. Understanding the order of operations (PEMDAS/BODMAS) is crucial in solving such expressions accurately. PEMDAS stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). BODMAS stands for Brackets, Orders, Division and Multiplication (from left to right), and Addition and Subtraction (from left to right). Both acronyms represent the same hierarchy of operations.

To effectively tackle this expression, we will divide it into three main parts: the numerator's numerator, the numerator's denominator, and the final division. This approach will allow us to manage the complexity and reduce the chances of error. First, we'll focus on the innermost operations within the parentheses and brackets, following the order of operations. This involves calculating the interest rate per period and the total number of periods. Then, we'll move on to the exponentiation, which requires careful handling of the negative exponent. After that, we'll perform the subtraction within the brackets and the multiplication in the numerator. Finally, we'll perform the divisions to arrive at the ultimate result. Each step will be meticulously explained to ensure clarity and comprehension. The use of intermediate results will also help in verifying the accuracy of our calculations.

Let's begin by calculating the numerator's numerator. This part of the expression is $11,000(\frac{0.09}{12})$. To solve this, we first divide 0.09 by 12, which gives us 0.0075. Then, we multiply 11,000 by 0.0075. This calculation yields 82.5. Next, we will move on to calculating the denominator of the numerator. This part is more complex and involves several steps. The expression is $[1-(1+\frac{0.09}{12})^{-12 \times 6}]$. First, we calculate $\frac{0.09}{12}$, which is 0.0075. Then, we add 1 to 0.0075, resulting in 1.0075. Now, we need to calculate the exponent, which is $-12 \times 6$, giving us -72. So, we have $(1.0075)^{-72}$. This means we need to calculate the reciprocal of $(1.0075)^{72}$. Using a calculator, we find that $(1.0075)^{72}$ is approximately 1.71255277. Therefore, $(1.0075)^{-72}$ is approximately $\frac{1}{1.71255277}$, which is about 0.58399188. Next, we subtract this result from 1, giving us $1 - 0.58399188$, which is approximately 0.41600812. Now that we have the numerator's numerator (82.5) and the numerator's denominator (0.41600812), we can divide the numerator by the denominator. So, $\frac{82.5}{0.41600812}$ is approximately 198.31231. Finally, we divide this result by 198.28, as per the original expression. Therefore, $\frac{198.31231}{198.28}$ is approximately 1.000163. This completes our step-by-step calculation.

Focusing on the numerator of the main expression, $\frac{11,000(\frac{0.09}{12})}{[1-(1+\frac{0.09}{12})^{-12 \times 6}]}$, it's crucial to dissect each component for clarity. The numerator itself comprises two primary parts: the initial product of 11,000 and the fraction $\frac{0.09}{12}$, and the more complex denominator involving exponentiation and subtraction. First, we address the simpler part. Dividing 0.09 by 12, we obtain 0.0075. Multiplying this result by 11,000 gives us 82.5. This intermediate result will be used later. The second, more intricate part of the numerator's calculation lies in the denominator: $[1-(1+\frac{0.09}{12})^{-12 \times 6}]$. To untangle this, we follow the order of operations meticulously. Starting from the innermost operations, we again encounter the fraction $\frac{0.09}{12}$, which we've already established is 0.0075. Adding 1 to this value results in 1.0075. The next step involves handling the exponent. The exponent is $-12 \times 6$, which equals -72. This means we need to calculate $(1.0075)^{-72}$, which is equivalent to finding the reciprocal of $(1.0075)^{72}$. Calculating $(1.0075)^{72}$ requires the use of a calculator or computational tool. The result is approximately 1.71255277. Taking the reciprocal, we get $\frac{1}{1.71255277}$, which is approximately 0.58399188. Finally, we subtract this value from 1: $1 - 0.58399188$, which equals approximately 0.41600812. Therefore, the denominator of the numerator simplifies to 0.41600812. Now we have both parts of the numerator calculated: 82.5 and 0.41600812. Dividing 82.5 by 0.41600812 gives us approximately 198.31231. This completes the detailed breakdown of the numerator, providing a clear pathway through the calculations.

In the given expression, the final division involves dividing the result obtained from the numerator by the value 198.28. This value, 198.28, acts as the ultimate denominator in our calculation. While it might seem straightforward, understanding its role in the broader context is crucial. The entire expression can be viewed as a fraction where the complex numerator we've meticulously calculated is divided by this single value. So, after all the intricate calculations in the numerator, the final step is to divide that result by 198.28. This step essentially scales the result of the numerator calculations down or up, depending on the magnitude of the numerator's result relative to 198.28. In our case, the numerator's result was approximately 198.31231. Dividing this by 198.28, we get a final result that is very close to 1. This highlights the importance of each step in the calculation, as even slight variations in intermediate results can impact the final answer. The denominator, 198.28, serves as a fixed point against which the numerator's value is compared, giving us the final scaled result. There isn't any further breakdown needed for 198.28 as it’s a constant value in the expression. Its significance lies in its role as the final divisor, shaping the ultimate outcome of the calculation. Understanding this simple yet crucial role helps in appreciating the overall structure and flow of the mathematical expression. This final division provides context to the complex calculations performed earlier, giving a clear and concise answer to the problem.

Having meticulously calculated both the numerator and the denominator, the final step is to divide the numerator's result by the given denominator, 198.28. From our previous calculations, the numerator simplifies to approximately 198.31231. Now, we perform the final division: $\frac{198.31231}{198.28}$. This division yields a result of approximately 1.000163. This value is very close to 1, indicating that the numerator and the denominator are almost equal. The slight difference arises from the cumulative effect of rounding errors in the intermediate calculations. While we've used approximations at various stages to simplify the process, these approximations can introduce minor discrepancies in the final result. However, the result of 1.000163 provides a highly accurate answer to the complex expression. To summarize, we started with a seemingly daunting expression involving multiple operations, broke it down into manageable parts, calculated each part systematically, and then combined the results to arrive at the final answer. The key to solving such expressions lies in understanding the order of operations and meticulously executing each step. The final result, approximately 1.000163, completes the calculation and provides a clear and concise solution to the problem. This exercise demonstrates the importance of precision and attention to detail in mathematical calculations, especially when dealing with complex expressions. The step-by-step approach ensures accuracy and helps in understanding the underlying mathematical principles involved.

In conclusion, we have successfully calculated the value of the complex mathematical expression $\frac{\frac{11,000(\frac{0.09}{12})}{[1-(1+\frac{0.09}{12})^{-12 \times 6}]}}{198.28}$. By breaking down the expression into smaller, manageable parts and meticulously performing each calculation, we arrived at the final result of approximately 1.000163. This process highlighted the importance of adhering to the order of operations (PEMDAS/BODMAS) and the significance of precision in mathematical calculations. The step-by-step approach not only ensures accuracy but also enhances understanding of the underlying mathematical principles. Each component of the expression, from the initial multiplication and division to the more complex exponentiation and subtraction, was carefully evaluated and calculated. The use of intermediate results allowed for verification and minimized the chances of error. The final division by 198.28 provided the scaling factor that led to the ultimate solution. This exercise serves as a practical demonstration of how complex mathematical problems can be solved by systematic decomposition and careful execution. The resulting value, 1.000163, represents the culmination of all the individual calculations and provides a clear and concise answer to the posed mathematical challenge. Understanding such processes is crucial in various fields, including finance, engineering, and data analysis, where complex calculations are frequently encountered. The ability to break down and solve intricate expressions is a valuable skill that can be applied across diverse domains.