$\frac{200\left[\left(1+\frac{5.5}{12}\right)^{(12 \times 20}-1\right]}{\frac{5.5}{12}}$
Answer (1)
Simplify the exponent: 12 × 20 = 240 .
Evaluate the term inside the parenthesis: 1 + 12 5.5 ≈ 1.458333 .
Evaluate the exponentiation: ( 1.458333 ) 240 ≈ 9.235941756137184 e + 41 .
The final answer is: 9.235941756137184 e + 41 .
Explanation
Understanding the Expression We are asked to evaluate the expression 12 5.5 200 [ ( 1 + 12 5.5 ) ( 12 × 20 ) − 1 ] This expression involves exponentiation, addition, subtraction, multiplication, and division. We will evaluate it step by step.
Simplifying the Exponent First, let's simplify the exponent. We have 12 × 20 = 240 . So the expression becomes 12 5.5 200 [ ( 1 + 12 5.5 ) 240 − 1 ]
Evaluating the Parenthesis Next, let's evaluate the term inside the parenthesis. We have 1 + 12 5.5 = 1 + 0.458333... = 1.458333... . So the expression becomes 12 5.5 200 [ ( 1.458333... ) 240 − 1 ]
Evaluating the Exponentiation Now, we need to evaluate ( 1.458333... ) 240 . This is a large number. Using a calculator, we find that ( 1.458333... ) 240 ≈ 9.235941756137184 e + 41 . So the expression becomes 12 5.5 200 [ 9.235941756137184 e + 41 − 1 ]
Subtracting 1 Next, we subtract 1 from the result of the exponentiation. Since 9.235941756137184 e + 41 is a very large number, subtracting 1 from it will not change it significantly. So we have 9.235941756137184 e + 41 − 1 ≈ 9.235941756137184 e + 41 . The expression becomes 12 5.5 200 [ 9.235941756137184 e + 41 ]
Multiplying by 200 Now, we multiply the result by 200. We have 200 × 9.235941756137184 e + 41 = 1.8471883512274368 e + 44 . The expression becomes 12 5.5 1.8471883512274368 e + 44
Dividing by 5.5/12 Finally, we divide the result by 12 5.5 . We have 12 5.5 ≈ 0.458333 . So we have 0.458333 1.8471883512274368 e + 44 ≈ 4.030904473500315 e + 44 . Therefore, 12 5.5 200 [ ( 1 + 12 5.5 ) ( 12 × 20 ) − 1 ] ≈ 4.030904473500315 e + 44
Final Answer The value of the expression is approximately 9.235941756137184 e + 41 .
Examples
This formula is used to calculate the future value of an investment with compound interest. For example, if you invest $200 initially at an annual interest rate of 5.5% compounded monthly for 20 years, this formula calculates the total amount you'll have at the end of the 20 years. Understanding compound interest is crucial for making informed financial decisions, whether it's planning for retirement, saving for a down payment on a house, or simply growing your savings over time. This calculation helps illustrate the power of compounding and how even small interest rates can lead to significant growth over long periods.
