Find the Conjugate + Solve (1) \( \frac{6+\sqrt{81}}{2+\sqrt{81}} \) (2) \( \frac{2+\sqrt{2}}{7-\sqrt{2}} \) (3) \( 37+\sqrt{7} \) \( 20+\sqrt{7} \)

Calculate the value by following steps: - step0: Calculate: \(37+\sqrt{7}\) Calculate or simplify the expression \( 20+\sqrt{7} \). Calculate the value by following steps: - step0: Calculate: \(20+\sqrt{7}\) Calculate or simplify the expression \( (6+\sqrt{81})/(2+\sqrt{81}) \). Calculate the value by following steps: - step0: Calculate: \(\frac{\left(6+\sqrt{81}\right)}{\left(2+\sqrt{81}\right)}\) - step1: Remove the parentheses: \(\frac{6+\sqrt{81}}{2+\sqrt{81}}\) - step2: Simplify the root: \(\frac{6+9}{2+\sqrt{81}}\) - step3: Simplify the root: \(\frac{6+9}{2+9}\) - step4: Add the numbers: \(\frac{15}{2+9}\) - step5: Add the numbers: \(\frac{15}{11}\) Calculate or simplify the expression \( (2+\sqrt{2})/(7-\sqrt{2}) \). Calculate the value by following steps: - step0: Calculate: \(\frac{\left(2+\sqrt{2}\right)}{\left(7-\sqrt{2}\right)}\) - step1: Remove the parentheses: \(\frac{2+\sqrt{2}}{7-\sqrt{2}}\) - step2: Multiply by the Conjugate: \(\frac{\left(2+\sqrt{2}\right)\left(7+\sqrt{2}\right)}{\left(7-\sqrt{2}\right)\left(7+\sqrt{2}\right)}\) - step3: Multiply the numbers: \(\frac{\left(2+\sqrt{2}\right)\left(7+\sqrt{2}\right)}{47}\) - step4: Multiply the numbers: \(\frac{16+9\sqrt{2}}{47}\) Let's solve each expression step by step. ### (1) \( \frac{6+\sqrt{81}}{2+\sqrt{81}} \) First, we simplify the expression: - Calculate \( \sqrt{81} = 9 \). - Substitute this value into the expression: \[ \frac{6 + 9}{2 + 9} = \frac{15}{11} \] Thus, the simplified result is: \[ \frac{15}{11} \quad \text{or} \quad 1\frac{4}{11} \quad \text{or} \quad 1.3636\ldots \] ### (2) \( \frac{2+\sqrt{2}}{7-\sqrt{2}} \) Next, we simplify this expression by multiplying the numerator and denominator by the conjugate of the denominator: - The conjugate of \( 7 - \sqrt{2} \) is \( 7 + \sqrt{2} \). Now, we perform the multiplication: \[ \frac{(2+\sqrt{2})(7+\sqrt{2})}{(7-\sqrt{2})(7+\sqrt{2})} \] Calculating the denominator: \[ (7-\sqrt{2})(7+\sqrt{2}) = 49 - 2 = 47 \] Calculating the numerator: \[ (2+\sqrt{2})(7+\sqrt{2}) = 14 + 2\sqrt{2} + 7\sqrt{2} + 2 = 16 + 9\sqrt{2} \] Putting it all together, we have: \[ \frac{16 + 9\sqrt{2}}{47} \] ### (3) \( 37+\sqrt{7} \) This expression is already in its simplest form. However, we can also provide a numerical approximation: \[ 37 + \sqrt{7} \approx 39.645751 \] ### (4) \( 20+\sqrt{7} \) Similarly, this expression is also in its simplest form, with a numerical approximation: \[ 20 + \sqrt{7} \approx 22.645751 \] ### Summary of Results 1. \( \frac{6+\sqrt{81}}{2+\sqrt{81}} = \frac{15}{11} \) 2. \( \frac{2+\sqrt{2}}{7-\sqrt{2}} = \frac{16 + 9\sqrt{2}}{47} \) 3. \( 37+\sqrt{7} \approx 39.645751 \) 4. \( 20+\sqrt{7} \approx 22.645751 \)