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: $$\sqrt[3]{xy^{2}} = \sqrt[3]{x} \cdot \sqrt[3]{y^{2}}$$. 3. **Rewrite the expression:** $$\frac{x}{\sqrt[3]{x} \cdot \sqrt[3]{y^{2}}} = \frac{x}{x^{\frac{1}{3}} y^{\frac{2}{3}}}$$ 4. **Express numerator and denominator with exponents:** $$x = x^{1}$$ 5. **Divide powers of x:** $$\frac{x^{1}}{x^{\frac{1}{3}}} = x^{1 - \frac{1}{3}} = x^{\frac{2}{3}}$$ 6. **Rewrite the entire expression:** $$\frac{x}{\sqrt[3]{xy^{2}}} = \frac{x^{1}}{x^{\frac{1}{3}} y^{\frac{2}{3}}} = \frac{x^{\frac{2}{3}}}{y^{\frac{2}{3}}}$$ 7. **Express as a single fraction:** $$= \left(\frac{x}{y}\right)^{\frac{2}{3}}$$ **Final answer:** $$\boxed{\left(\frac{x}{y}\right)^{\frac{2}{3}}}$$