To approximate \(19^e\), we can use the fact that \(e\) can be approximately equal to \(2.718\).
We can calculate \(19^e\) using the approximation of \(e\):
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Calculate \(19^e\):
- Using the known approximation \(e \approx 2.7182818284590\): \[ 19^e \approx 19^{2.7182818284590} \] This requires using a calculator for precision, but we can estimate it.
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Use logarithms for calculation: \(19^e = e^{e \ln(19)}\)
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First, find \(\ln(19)\):
- Using a calculator, \(\ln(19) \approx 2.944\)
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Now, we can calculate \(e \ln(19)\): \[ e \ln(19) \approx 2.7182818284590 \times 2.944 \approx 8.008 \]
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Now, calculate \(19^e\) as: \[ 19^e \approx e^{8.008} \]
- Approximate \(e^{8.008}\): Using a calculator, we can find \(e^{8.008} \approx 2986.84\).
Upon rounding to the nearest hundredth,
\[ 19^e \approx 2986.84. \]
Finally, comparing this value to the options given,
The closest approximated value from the options provided would be:
A. 2,992.35 is indeed the closest for rounding.