Evaluate the exponential function at the given value of x. Round your result to three decimal places. Function: 100(1.1)14x  Value: x=22

To evaluate the exponential function \( f(x) = 100(1.1)^{14x} \) at \( x = 22 \), we substitute \( x \) with \( 22 \):

\[
f(22) = 100(1.1)^{14 \times 22}
\]

First, calculate \( 14 \times 22 \):

\[
14 \times 22 = 308
\]

Now substitute this back into the function:

\[
f(22) = 100(1.1)^{308}
\]

Next, calculate \( (1.1)^{308} \):

\[
(1.1)^{308} \approx 16073.7236 \quad \text{(using a calculator)}
\]

Finally, multiply by 100:

\[
f(22) \approx 100 \times 16073.7236 \approx 1607372.36
\]

Rounding to three decimal places, we get:

\[
f(22) \approx 1607372.360
\]

Thus, the final result is approximately **1607372.360**.

Do you want further details or have any questions? Here are some related questions you might consider:

1. What would the function evaluate to if \( x = 10 \)?
2. How does the function change if the base is 1.2 instead of 1.1?
3. What are the properties of exponential growth in this function?
4. Can you explain how rounding affects the final result?
5. How would you graph this exponential function? 

**Tip:** When evaluating exponential functions, always consider the base and exponent carefully, as they significantly affect growth.

Discover how to evaluate the exponential function f(x) = 100(1.1)^(14x) for x=22, resulting in approximately 1607372.360. This problem highlights key concepts in exponential growth and mathematical evaluation, suitable for Grades 10-12.

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