1. **State the problem:** Simplify the expression $$\sqrt[3]{54x} - \sqrt[3]{16x}$$. 2. **Recall the cube root properties:** The cube root of a product is the product of the cube roots: $$\sqrt[3]{ab} = \sqrt[3]{a} \cdot \sqrt[3]{b}$$. 3. **Factor inside the cube roots:** - $$54x = 27 \cdot 2 \cdot x$$ - $$16x = 8 \cdot 2 \cdot x$$ 4. **Rewrite the cube roots using factors:** $$\sqrt[3]{54x} = \sqrt[3]{27 \cdot 2x} = \sqrt[3]{27} \cdot \sqrt[3]{2x} = 3 \cdot \sqrt[3]{2x}$$ $$\sqrt[3]{16x} = \sqrt[3]{8 \cdot 2x} = \sqrt[3]{8} \cdot \sqrt[3]{2x} = 2 \cdot \sqrt[3]{2x}$$ 5. **Substitute back:** $$3 \cdot \sqrt[3]{2x} - 2 \cdot \sqrt[3]{2x}$$ 6. **Factor out the common term:** $$\left(3 - 2\right) \cdot \sqrt[3]{2x} = 1 \cdot \sqrt[3]{2x} = \sqrt[3]{2x}$$ **Final answer:** $$\sqrt[3]{2x}$$