1. **State the problem:** Simplify the expression $$\sqrt[3]{50} + \sqrt[3]{75}$$. 2. **Recall the cube root properties:** The cube root of a product is the product of the cube roots: $$\sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b}$$. 3. **Factor each radicand to extract perfect cubes:** $$50 = 25 \times 2 = 5^2 \times 2$$ $$75 = 25 \times 3 = 5^2 \times 3$$ 4. **Rewrite the cube roots:** $$\sqrt[3]{50} = \sqrt[3]{5^2 \times 2}$$ $$\sqrt[3]{75} = \sqrt[3]{5^2 \times 3}$$ 5. **Since 5^2 is not a perfect cube, we cannot simplify further by extracting cubes.** 6. **Combine the cube roots if possible:** $$\sqrt[3]{50} + \sqrt[3]{75} = \sqrt[3]{5^2 \times 2} + \sqrt[3]{5^2 \times 3}$$ 7. **Factor out the common cube root of $$5^2$$:** $$= \sqrt[3]{5^2} (\sqrt[3]{2} + \sqrt[3]{3})$$ 8. **Final simplified form:** $$= \sqrt[3]{25} (\sqrt[3]{2} + \sqrt[3]{3})$$ This is the simplest exact form. **Answer:** $$\sqrt[3]{25} (\sqrt[3]{2} + \sqrt[3]{3})$$