To estimate \( 3^{\sqrt{29}} \) correct to two decimal places, we first need to find an approximation for \( \sqrt{29} \).
The square root of 29 is approximately \( 5.385 \). However, since we are looking for an approximation with the least number of decimal places, we can round it to one decimal place:
\[ \sqrt{29} \approx 5.4 \]
Next, we need to compute \( 3^{\sqrt{29}} \) using \( 5.4 \):
\[ 3^{5.4} \approx 3^{5} \times 3^{0.4} \]
Calculating \( 3^5 = 243 \).
Now we need to estimate \( 3^{0.4} \). We can approximate \( 3^{0.4} \) using the exponential function or possibly find it with a calculator, but for the purposes of this question, we can also observe that \( 3^{0.4} \) is approximately \( 1.514 \) (you can use \( 3^{0.5} = \sqrt{3} \approx 1.732 \) and some knowledge of exponents).
From this approximation, we can multiply:
\[ 3^{5.4} \approx 243 \times 1.514 \approx 368.382 \]
In order to match the options presented, we round \( 368.382 \) to two decimal places, resulting in approximately \( 368.38 \).
The options presented are:
A. 371.00
B. 5.385
C. 5.3851648
D. 538516
None of these correspond directly to our rounded estimate. However, the only option that could relate logically and provide some correct information about \( 3^{\sqrt{29}} \) is the one that mentions \( 5.385 \) (as it represents a close approximation of \( \sqrt{29} \)), making it useful in calculating the power estimation.
Thus, the best answer in context is:
B. \( 5.385 \)
This is because it gives us a straightforward approximation to apply in calculations, without unnecessary precision that would not significantly affect the two-decimal-place accuracy we are trying to achieve.
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