Evaluate the following expression for kequals4 and nequals8.

18 comma 000 left parenthesis 1 plus StartFraction 0.10 Over k EndFraction right parenthesis Superscript kn
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Part 1
18 comma 000 left parenthesis 1 plus StartFraction 0.10 Over k EndFraction right parenthesis Superscript kn
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​(Do not round until the final answer. Then round to two decimal places as​ needed.)

1 answer

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:

  1. Substitute \( k \) and \( n \) into the expression.
  2. Calculate \( \frac{0.10}{k} \).
  3. Solve for the power \( kn \).
  4. 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:

\[ \boxed{39,744.72} \]