To evaluate the expression \( 18,000 \left( 1 + \frac{0.10}{k} \right)^{kn} \) for \( k = 4 \) and \( n = 8 \), we will follow these steps:
- Substitute \( k \) and \( n \) into the expression.
- Calculate \( \frac{0.10}{k} \).
- Solve for the power \( kn \).
- Evaluate the entire expression.
Let's go through these steps:
Step 1: Substitute \( k \) and \( n \)
Given \( k = 4 \) and \( n = 8 \):
\[ kn = 4 \times 8 = 32 \]
Step 2: Calculate \( \frac{0.10}{k} \)
Now, substituting \( k \):
\[ \frac{0.10}{k} = \frac{0.10}{4} = 0.025 \]
Step 3: Substitute into the expression
Now we substitute back into the original expression:
\[ 18,000 \left( 1 + 0.025 \right)^{32} \]
This simplifies to:
\[ 18,000 \left( 1.025 \right)^{32} \]
Step 4: Calculate \( (1.025)^{32} \)
Now, we will calculate \( (1.025)^{32} \):
\[ (1.025)^{32} \approx 2.208040 \]
Step 5: Multiply by 18,000
Now, we multiply this value by 18,000:
\[ 18,000 \times 2.208040 \approx 39,744.72 \]
Final Answer
Thus, rounding to two decimal places, our final answer is:
\[ \boxed{39,744.72} \]