1. **State the problem:** Simplify the expression $$\frac{3 \sqrt{6} + 16}{\sqrt{6}}$$. 2. **Recall the rule:** When dividing terms by a square root, you can split the fraction into separate terms: $$\frac{a + b}{c} = \frac{a}{c} + \frac{b}{c}$$. 3. **Apply the rule:** $$\frac{3 \sqrt{6} + 16}{\sqrt{6}} = \frac{3 \sqrt{6}}{\sqrt{6}} + \frac{16}{\sqrt{6}}$$ 4. **Simplify the first term:** $$\frac{3 \sqrt{6}}{\sqrt{6}} = 3 \cancel{\frac{\sqrt{6}}{\sqrt{6}}} = 3$$ 5. **Simplify the second term:** $$\frac{16}{\sqrt{6}}$$ 6. **Rationalize the denominator:** Multiply numerator and denominator by $$\sqrt{6}$$ to remove the square root from the denominator: $$\frac{16}{\sqrt{6}} \times \frac{\sqrt{6}}{\sqrt{6}} = \frac{16 \sqrt{6}}{6}$$ 7. **Simplify the fraction:** $$\frac{16 \sqrt{6}}{6} = \frac{8 \sqrt{6}}{3}$$ 8. **Write the final simplified expression:** $$3 + \frac{8 \sqrt{6}}{3}$$ **Final answer:** $$3 + \frac{8 \sqrt{6}}{3}$$