1. **State the problem:** Simplify the expression $$-3(-162)^{\frac{1}{3}}$$ and match it to one of the given choices. 2. **Recall the cube root property:** For any real number $a$, $$a^{\frac{1}{3}} = \sqrt[3]{a}$$ which means the cube root of $a$. 3. **Simplify inside the cube root:** We need to find $$\sqrt[3]{-162}$$. 4. **Factor 162:** $$162 = 2 \times 81 = 2 \times 3^4$$. 5. **Rewrite the cube root:** $$\sqrt[3]{-162} = \sqrt[3]{-1 \times 2 \times 3^4} = \sqrt[3]{-1} \times \sqrt[3]{2} \times \sqrt[3]{3^4}$$. 6. **Simplify each cube root:** - $$\sqrt[3]{-1} = -1$$ - $$\sqrt[3]{2}$$ stays as is - $$\sqrt[3]{3^4} = \sqrt[3]{3^3 \times 3} = 3 \times \sqrt[3]{3}$$ 7. **Combine the terms:** $$\sqrt[3]{-162} = -1 \times \sqrt[3]{2} \times 3 \times \sqrt[3]{3} = -3 \times \sqrt[3]{2} \times \sqrt[3]{3} = -3 \times \sqrt[3]{2 \times 3} = -3 \times \sqrt[3]{6}$$ 8. **Multiply by -3 outside:** $$-3 \times (-3 \times \sqrt[3]{6}) = \cancel{-3} \times \cancel{-3} \times \sqrt[3]{6} = 9 \times \sqrt[3]{6}$$ 9. **Final answer:** $$9 \sqrt[3]{6}$$ This matches the choice: **9 ∛6**.