9. \( \frac{3}{6-\sqrt{2}} \)

To simplify \( \frac{3}{6-\sqrt{2}} \), you can multiply the numerator and the denominator by the conjugate of the denominator, \( 6+\sqrt{2} \). This gives you: \[ \frac{3(6+\sqrt{2})}{(6-\sqrt{2})(6+\sqrt{2})} = \frac{18 + 3\sqrt{2}}{36 - 2} = \frac{18 + 3\sqrt{2}}{34} = \frac{9 + \frac{3\sqrt{2}}{2}}{17} \] Now you've got a nice simplified form! Additionally, the technique of multiplying by the conjugate is widely used in algebra to rationalize denominators. This helps in simplifying expressions, making calculations easier, and avoiding complex fractions in equations. Plus, this method can often reveal relationships or structures within mathematical problems that are not immediately obvious!