Use the function \( f(x)=3^{x} \) to find the approximation \( 3^{\sqrt{3}} \), correct to three decimal places, by creating a table with \( x \)-values closer and closer to \( \sqrt{3} \) and the corresponding \( f(x) \) values. (4 points)

To approximate \( 3^{\sqrt{3}} \) using the function \( f(x) = 3^x \), we will create a table with \( x \)-values that are progressively closer to \( \sqrt{3} \). First, we need to find the approximate value of \( \sqrt{3} \): \[ \sqrt{3} \approx 1.732 \] Next, we will choose \( x \)-values around \( 1.732 \) and calculate \( f(x) = 3^x \) for those values. We can start with values like \( 1.7 \), \( 1.71 \), \( 1.72 \), \( 1.73 \), and \( 1.74 \). Let's calculate \( f(x) \) for these \( x \)-values: 1. \( x = 1.7 \) 2. \( x = 1.71 \) 3. \( x = 1.72 \) 4. \( x = 1.73 \) 5. \( x = 1.74 \) Now, I will calculate \( f(x) \) for these values. Calculate the value by following steps: - step0: Calculate: \(3^{1.72}\) - step1: Convert the expressions: \(3^{\frac{43}{25}}\) - step2: Transform the expression: \(\sqrt[25]{3^{43}}\) - step3: Rewrite the expression: \(\sqrt[25]{3^{25}\times 3^{18}}\) - step4: Rewrite the expression: \(\sqrt[25]{3^{25}}\times \sqrt[25]{3^{18}}\) - step5: Rewrite the expression: \(3\sqrt[25]{3^{18}}\) Calculate or simplify the expression \( 3^(1.74) \). Calculate the value by following steps: - step0: Calculate: \(3^{1.74}\) - step1: Convert the expressions: \(3^{\frac{87}{50}}\) - step2: Transform the expression: \(\sqrt[50]{3^{87}}\) - step3: Rewrite the expression: \(\sqrt[50]{3^{50}\times 3^{37}}\) - step4: Rewrite the expression: \(\sqrt[50]{3^{50}}\times \sqrt[50]{3^{37}}\) - step5: Rewrite the expression: \(3\sqrt[50]{3^{37}}\) Calculate or simplify the expression \( 3^(1.71) \). Calculate the value by following steps: - step0: Calculate: \(3^{1.71}\) - step1: Convert the expressions: \(3^{\frac{171}{100}}\) - step2: Transform the expression: \(\sqrt[100]{3^{171}}\) - step3: Rewrite the expression: \(\sqrt[100]{3^{100}\times 3^{71}}\) - step4: Rewrite the expression: \(\sqrt[100]{3^{100}}\times \sqrt[100]{3^{71}}\) - step5: Rewrite the expression: \(3\sqrt[100]{3^{71}}\) Calculate or simplify the expression \( 3^(1.73) \). Calculate the value by following steps: - step0: Calculate: \(3^{1.73}\) - step1: Convert the expressions: \(3^{\frac{173}{100}}\) - step2: Transform the expression: \(\sqrt[100]{3^{173}}\) - step3: Rewrite the expression: \(\sqrt[100]{3^{100}\times 3^{73}}\) - step4: Rewrite the expression: \(\sqrt[100]{3^{100}}\times \sqrt[100]{3^{73}}\) - step5: Rewrite the expression: \(3\sqrt[100]{3^{73}}\) Calculate or simplify the expression \( 3^(1.7) \). Calculate the value by following steps: - step0: Calculate: \(3^{1.7}\) - step1: Convert the expressions: \(3^{\frac{17}{10}}\) - step2: Transform the expression: \(\sqrt[10]{3^{17}}\) - step3: Rewrite the expression: \(\sqrt[10]{3^{10}\times 3^{7}}\) - step4: Rewrite the expression: \(\sqrt[10]{3^{10}}\times \sqrt[10]{3^{7}}\) - step5: Rewrite the expression: \(3\sqrt[10]{3^{7}}\) - step6: Calculate: \(3\sqrt[10]{2187}\) Here are the calculated values of \( f(x) = 3^x \) for the chosen \( x \)-values: | \( x \) | \( f(x) = 3^x \) | |-----------|---------------------| | 1.7 | 6.473 | | 1.71 | 6.545 | | 1.72 | 6.617 | | 1.73 | 6.690 | | 1.74 | 6.764 | Now, we can observe the values of \( f(x) \) as \( x \) approaches \( \sqrt{3} \approx 1.732 \). To find a more precise approximation of \( 3^{\sqrt{3}} \), we can take the average of the values at \( x = 1.72 \) and \( x = 1.73 \): \[ \text{Average} = \frac{6.617 + 6.690}{2} \approx 6.6535 \] Thus, the approximation of \( 3^{\sqrt{3}} \) correct to three decimal places is: \[ \boxed{6.654} \]