1. **State the problem:** Simplify the expression $$7\sqrt[3]{2x} - 3\sqrt[3]{16x} - 3\sqrt[3]{8x}$$. 2. **Rewrite cube roots with prime factors:** - $$\sqrt[3]{16x} = \sqrt[3]{2^4 x} = \sqrt[3]{2^3 \cdot 2 x} = 2\sqrt[3]{2x}$$ - $$\sqrt[3]{8x} = \sqrt[3]{2^3 x} = 2\sqrt[3]{x}$$ 3. **Substitute back:** $$7\sqrt[3]{2x} - 3(2\sqrt[3]{2x}) - 3(2\sqrt[3]{x}) = 7\sqrt[3]{2x} - 6\sqrt[3]{2x} - 6\sqrt[3]{x}$$ 4. **Combine like terms:** $$ (7 - 6)\sqrt[3]{2x} - 6\sqrt[3]{x} = \sqrt[3]{2x} - 6\sqrt[3]{x}$$ 5. **Final simplified form:** $$\boxed{\sqrt[3]{2x} - 6\sqrt[3]{x}}$$