A) Approximate \( 11^{\sqrt{31}} \), correct to one decimal place, by using a sequence of values that gets closer and closer to \( \sqrt{31} \). (1 point)

To approximate \( 11^{\sqrt{31}} \) using a sequence of values that gets closer to \( \sqrt{31} \), we can start by estimating \( \sqrt{31} \). 1. **Estimate \( \sqrt{31} \)**: - We know that \( 5^2 = 25 \) and \( 6^2 = 36 \), so \( 5 < \sqrt{31} < 6 \). - We can refine our estimate by trying the midpoint: \( 5.5^2 = 30.25 \) (too low) and \( 5.6^2 = 31.36 \) (too high). - Thus, \( 5.5 < \sqrt{31} < 5.6 \). 2. **Narrowing down further**: - Let's try \( 5.57^2 = 31.0649 \) (too high) and \( 5.56^2 = 30.9136 \) (too low). - So, \( 5.56 < \sqrt{31} < 5.57 \). 3. **Using a more precise value**: - We can take \( \sqrt{31} \approx 5.57 \) for our calculations. Now we can calculate \( 11^{\sqrt{31}} \) using \( \sqrt{31} \approx 5.57 \): \[ 11^{\sqrt{31}} \approx 11^{5.57} \] Next, we will calculate \( 11^{5.57} \) to one decimal place. Let's perform the calculation. Calculate the value by following steps: - step0: Calculate: \(11^{5.57}\) - step1: Convert the expressions: \(11^{\frac{557}{100}}\) - step2: Transform the expression: \(\sqrt[100]{11^{557}}\) - step3: Rewrite the expression: \(\sqrt[100]{11^{500}\times 11^{57}}\) - step4: Rewrite the expression: \(\sqrt[100]{11^{500}}\times \sqrt[100]{11^{57}}\) - step5: Rewrite the expression: \(11^{5}\sqrt[100]{11^{57}}\) The calculation gives us: \[ 11^{5.57} \approx 631767.464488 \] Rounding this to one decimal place, we get: \[ 11^{\sqrt{31}} \approx 631767.5 \] Thus, the approximate value of \( 11^{\sqrt{31}} \), correct to one decimal place, is \( 631767.5 \).