18. \( \frac{6+5 \sqrt{2}}{4+3 \sqrt{2}} \)

To simplify \( \frac{6+5 \sqrt{2}}{4+3 \sqrt{2}} \), we can multiply the numerator and the denominator by the conjugate of the denominator, which is \( 4-3 \sqrt{2} \). This gives us: \[ \frac{(6+5\sqrt{2})(4-3\sqrt{2})}{(4+3\sqrt{2})(4-3\sqrt{2})}. \] Calculating the denominator first: \[ (4+3\sqrt{2})(4-3\sqrt{2}) = 4^2 - (3\sqrt{2})^2 = 16 - 18 = -2. \] Now for the numerator: \[ (6 + 5\sqrt{2})(4 - 3\sqrt{2}) = 6 \cdot 4 + 6 \cdot (-3\sqrt{2}) + 5\sqrt{2} \cdot 4 + 5\sqrt{2} \cdot (-3\sqrt{2}). \] Calculating each term: \[ 24 - 18\sqrt{2} + 20\sqrt{2} - 15 \cdot 2 = 24 + 2\sqrt{2} - 30 = -6 + 2\sqrt{2}. \] Putting it all together: \[ \frac{-6 + 2\sqrt{2}}{-2} = \frac{-6}{-2} + \frac{2\sqrt{2}}{-2} = 3 - \sqrt{2}. \] Thus, the simplified expression is: \[ \boxed{3 - \sqrt{2}}. \]