Determine the most simplified version of the expression below: \( \frac{7+2 \sqrt{5}}{7-2 \sqrt{5}} \) (Example: \( \frac{7-3 \sqrt{6}}{10} \) would be typed in as \( 7+-3 \mathrm{rad6} / 10 \) ) \( \square \)

To simplify the expression \( \frac{7+2 \sqrt{5}}{7-2 \sqrt{5}} \), we can multiply the numerator and the denominator by the conjugate of the denominator, which is \( 7 + 2 \sqrt{5} \): \[ \frac{(7+2\sqrt{5})(7+2\sqrt{5})}{(7-2\sqrt{5})(7+2\sqrt{5})} \] Calculating the denominator, we have: \[ (7-2\sqrt{5})(7+2\sqrt{5}) = 7^2 - (2\sqrt{5})^2 = 49 - 20 = 29 \] For the numerator, we expand \( (7+2\sqrt{5})^2 \): \[ (7+2\sqrt{5})^2 = 7^2 + 2(7)(2\sqrt{5}) + (2\sqrt{5})^2 = 49 + 28\sqrt{5} + 20 = 69 + 28\sqrt{5} \] Putting it all together: \[ \frac{69 + 28\sqrt{5}}{29} \] Now, we can split it into two fractions: \[ \frac{69}{29} + \frac{28\sqrt{5}}{29} = 2.379 + 0.966\sqrt{5} \] However, since we want it in the specified format, we express it as: \[ 2.379 + 0.966\mathrm{rad5} / 1 \] Therefore, the most simplified version is: \[ \frac{69 + 28\mathrm{rad5}}{29} \] Thus, the final output should be: \( 69+28 \mathrm{rad5} / 29 \)