1. **State the problem:** Rationalise the denominator of $$\frac{3\sqrt{5}}{\sqrt{6}}$$ and simplify the expression. 2. **Formula and rule:** To rationalise a denominator containing a square root, multiply numerator and denominator by the conjugate or the same root to eliminate the root in the denominator. 3. **Multiply numerator and denominator by $$\sqrt{6}$$:** $$\frac{3\sqrt{5}}{\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}} = \frac{3\sqrt{5} \times \sqrt{6}}{\sqrt{6} \times \sqrt{6}}$$ 4. **Simplify denominator:** $$\sqrt{6} \times \sqrt{6} = 6$$ 5. **Simplify numerator:** $$3 \times \sqrt{5 \times 6} = 3 \times \sqrt{30}$$ 6. **Final simplified expression:** $$\frac{3\sqrt{30}}{6}$$ 7. **Simplify the fraction:** $$\frac{3}{6} = \frac{1}{2}$$ 8. **Answer:** $$\frac{\sqrt{30}}{2}$$ This is the simplest form with a rationalised denominator.