1. **State the problem:** Simplify the expression $$\sqrt[5]{32y^{25}}$$. 2. **Recall the formula:** The fifth root of a product is the product of the fifth roots: $$\sqrt[5]{ab} = \sqrt[5]{a} \times \sqrt[5]{b}$$. 3. **Apply the rule to each part:** - For the number 32, note that $$32 = 2^5$$. - For the variable part, $$y^{25}$$, use the rule $$\sqrt[n]{x^m} = x^{\frac{m}{n}}$$. 4. **Simplify each part:** - $$\sqrt[5]{32} = \sqrt[5]{2^5} = 2$$. - $$\sqrt[5]{y^{25}} = y^{\frac{25}{5}} = y^5$$. 5. **Combine the results:** $$\sqrt[5]{32y^{25}} = 2y^5$$. **Final answer:** $$2y^5$$