1. **State the problem:** Simplify the expression $$4 \sqrt[3]{81} - 2 \sqrt[3]{72} - \sqrt[3]{24}$$ by reducing each radical and then combining like terms. 2. **Recall the properties of radicals:** - The cube root of a product is the product of the cube roots: $$\sqrt[3]{a \times b} = \sqrt[3]{a} \times \sqrt[3]{b}$$ - Simplify radicals by factoring out perfect cubes. 3. **Simplify each cube root:** - $$\sqrt[3]{81} = \sqrt[3]{27 \times 3} = \sqrt[3]{27} \times \sqrt[3]{3} = 3 \sqrt[3]{3}$$ - $$\sqrt[3]{72} = \sqrt[3]{9 \times 8} = \sqrt[3]{9} \times \sqrt[3]{8} = \sqrt[3]{9} \times 2 \sqrt[3]{1} = 2 \sqrt[3]{9}$$ but since $$9$$ is not a perfect cube, keep as is for now. - Actually, $$\sqrt[3]{72} = \sqrt[3]{8 \times 9} = 2 \sqrt[3]{9}$$ - $$\sqrt[3]{24} = \sqrt[3]{8 \times 3} = 2 \sqrt[3]{3}$$ 4. **Rewrite the expression substituting simplified radicals:** $$4 \times 3 \sqrt[3]{3} - 2 \times 2 \sqrt[3]{9} - 2 \sqrt[3]{3}$$ 5. **Calculate coefficients:** $$12 \sqrt[3]{3} - 4 \sqrt[3]{9} - 2 \sqrt[3]{3}$$ 6. **Note that $$\sqrt[3]{9}$$ cannot be simplified further, so keep it as is. Group like terms:** $$ (12 \sqrt[3]{3} - 2 \sqrt[3]{3}) - 4 \sqrt[3]{9} = 10 \sqrt[3]{3} - 4 \sqrt[3]{9}$$ 7. **Final simplified expression:** $$\boxed{10 \sqrt[3]{3} - 4 \sqrt[3]{9}}$$