1. **State the problem:** Simplify the expression $$\frac{\sqrt{y^7} \times \sqrt[5]{y^2}}{\sqrt[5]{y^4}}$$. 2. **Rewrite roots as exponents:** Recall that $$\sqrt{y^7} = y^{\frac{7}{2}}$$, $$\sqrt[5]{y^2} = y^{\frac{2}{5}}$$, and $$\sqrt[5]{y^4} = y^{\frac{4}{5}}$$. 3. **Substitute these into the expression:** $$\frac{y^{\frac{7}{2}} \times y^{\frac{2}{5}}}{y^{\frac{4}{5}}}$$ 4. **Use exponent rules:** When multiplying with the same base, add exponents: $$y^{\frac{7}{2} + \frac{2}{5}}$$ When dividing with the same base, subtract exponents: $$y^{\left(\frac{7}{2} + \frac{2}{5}\right) - \frac{4}{5}}$$ 5. **Calculate the exponent:** Find a common denominator for $$\frac{7}{2}$$ and $$\frac{2}{5}$$, which is 10. $$\frac{7}{2} = \frac{35}{10}, \quad \frac{2}{5} = \frac{4}{10}$$ Add: $$\frac{35}{10} + \frac{4}{10} = \frac{39}{10}$$ Subtract $$\frac{4}{5} = \frac{8}{10}$$: $$\frac{39}{10} - \frac{8}{10} = \frac{31}{10}$$ 6. **Final simplified expression:** $$y^{\frac{31}{10}}$$ This is the simplified form of the original expression.