1. **State the problem:** Simplify the expression $$\frac{x}{\sqrt[3]{xy^{2}}}$$. 2. **Recall the formula and rules:** The cube root of a product can be written as the product of cube roots: $$\sqrt[3]{a b} = \sqrt[3]{a} \cdot \sqrt[3]{b}$$. 3. **Rewrite the denominator:** $$\sqrt[3]{xy^{2}} = \sqrt[3]{x} \cdot \sqrt[3]{y^{2}} = x^{\frac{1}{3}} y^{\frac{2}{3}}$$. 4. **Rewrite the entire expression:** $$\frac{x}{x^{\frac{1}{3}} y^{\frac{2}{3}}} = \frac{x^{1}}{x^{\frac{1}{3}} y^{\frac{2}{3}}}$$. 5. **Divide the powers of x:** $$= \frac{\cancel{x^{1}}}{\cancel{x^{\frac{1}{3}}}} \cdot \frac{1}{y^{\frac{2}{3}}} = x^{1 - \frac{1}{3}} y^{-\frac{2}{3}} = x^{\frac{2}{3}} y^{-\frac{2}{3}}$$. 6. **Rewrite with positive exponents:** $$= \frac{x^{\frac{2}{3}}}{y^{\frac{2}{3}}} = \left( \frac{x}{y} \right)^{\frac{2}{3}}$$. **Final answer:** $$\boxed{\left( \frac{x}{y} \right)^{\frac{2}{3}}}$$