1. **State the problem:** Simplify the expression $$\frac{1}{3\sqrt{9 + \sqrt{2}}}$$. 2. **Recall the formula and rules:** To simplify expressions with radicals in the denominator, we often rationalize the denominator by multiplying numerator and denominator by a conjugate or an expression that removes the radical. 3. **Identify the denominator:** The denominator is $$3\sqrt{9 + \sqrt{2}}$$. 4. **Rationalize the denominator:** Multiply numerator and denominator by $$\sqrt{9 - \sqrt{2}}$$ to use the difference of squares formula: $$\sqrt{a + b} \times \sqrt{a - b} = \sqrt{a^2 - b^2}$$. 5. **Perform the multiplication:** $$\frac{1}{3\sqrt{9 + \sqrt{2}}} \times \frac{\sqrt{9 - \sqrt{2}}}{\sqrt{9 - \sqrt{2}}} = \frac{\sqrt{9 - \sqrt{2}}}{3 \sqrt{(9 + \sqrt{2})(9 - \sqrt{2})}}$$ 6. **Simplify inside the square root in the denominator:** $$ (9 + \sqrt{2})(9 - \sqrt{2}) = 9^2 - (\sqrt{2})^2 = 81 - 2 = 79 $$ 7. **So the denominator becomes:** $$3 \sqrt{79}$$ 8. **Final simplified expression:** $$\frac{\sqrt{9 - \sqrt{2}}}{3 \sqrt{79}}$$ This is the simplified form with a rationalized denominator. **Answer:** $$\frac{\sqrt{9 - \sqrt{2}}}{3 \sqrt{79}}$$