1. **Problem:** Simplify the expression $$\sqrt[3]{64x^4}$$. 2. **Formula and rules:** The cube root of a product is the product of the cube roots: $$\sqrt[3]{a \cdot b} = \sqrt[3]{a} \cdot \sqrt[3]{b}$$. 3. **Step 1:** Factor inside the cube root: $$\sqrt[3]{64x^4} = \sqrt[3]{64} \cdot \sqrt[3]{x^4}$$ 4. **Step 2:** Simplify $$\sqrt[3]{64}$$. Since $$64 = 4^3$$, we have: $$\sqrt[3]{64} = 4$$ 5. **Step 3:** Simplify $$\sqrt[3]{x^4}$$. Write $$x^4 = x^3 \cdot x$$, so: $$\sqrt[3]{x^4} = \sqrt[3]{x^3 \cdot x} = \sqrt[3]{x^3} \cdot \sqrt[3]{x} = x \cdot \sqrt[3]{x}$$ 6. **Step 4:** Combine the results: $$4 \cdot x \cdot \sqrt[3]{x} = 4x \sqrt[3]{x}$$ **Final answer:** $$\boxed{4x \sqrt[3]{x}}$$