1. **State the problem:** Simplify the expression $$\frac{1}{3\sqrt[3]{4} + \sqrt{2}}$$. 2. **Recall the formula and rules:** To simplify expressions with roots in the denominator, we often rationalize the denominator by multiplying numerator and denominator by the conjugate or an expression that removes the roots. 3. **Identify the conjugate:** The denominator is $$3\sqrt[3]{4} + \sqrt{2}$$. Its conjugate is $$3\sqrt[3]{4} - \sqrt{2}$$. 4. **Multiply numerator and denominator by the conjugate:** $$\frac{1}{3\sqrt[3]{4} + \sqrt{2}} \times \frac{3\sqrt[3]{4} - \sqrt{2}}{3\sqrt[3]{4} - \sqrt{2}} = \frac{3\sqrt[3]{4} - \sqrt{2}}{(3\sqrt[3]{4})^2 - (\sqrt{2})^2}$$ 5. **Calculate the denominator:** $$ (3\sqrt[3]{4})^2 = 9 (\sqrt[3]{4})^2 = 9 \sqrt[3]{16} $$ $$ (\sqrt{2})^2 = 2 $$ So denominator is $$9 \sqrt[3]{16} - 2$$. 6. **Final simplified form:** $$\frac{3\sqrt[3]{4} - \sqrt{2}}{9 \sqrt[3]{16} - 2}$$ This is the simplified expression with a rationalized denominator. **Answer:** $$\frac{3\sqrt[3]{4} - \sqrt{2}}{9 \sqrt[3]{16} - 2}$$