1. **State the problem:** Simplify the expression $$4 \sqrt[3]{81x} - 3 \sqrt[3]{192x}$$ assuming all variables are positive. 2. **Recall the cube root properties:** - $$\sqrt[3]{a \cdot b} = \sqrt[3]{a} \cdot \sqrt[3]{b}$$ - Cube roots can be factored to extract perfect cubes. 3. **Factor inside the cube roots:** - $$81 = 27 \times 3$$, so $$\sqrt[3]{81x} = \sqrt[3]{27 \times 3x} = \sqrt[3]{27} \cdot \sqrt[3]{3x}$$ - $$192 = 64 \times 3$$, so $$\sqrt[3]{192x} = \sqrt[3]{64 \times 3x} = \sqrt[3]{64} \cdot \sqrt[3]{3x}$$ 4. **Evaluate cube roots of perfect cubes:** - $$\sqrt[3]{27} = 3$$ - $$\sqrt[3]{64} = 4$$ 5. **Rewrite the expression:** $$4 \times 3 \times \sqrt[3]{3x} - 3 \times 4 \times \sqrt[3]{3x} = 12 \sqrt[3]{3x} - 12 \sqrt[3]{3x}$$ 6. **Combine like terms:** $$12 \sqrt[3]{3x} - 12 \sqrt[3]{3x} = 0$$ 7. **Final answer:** $$0$$ The expression simplifies to zero.