1. **State the problem:** Simplify the expression $$\sqrt[3]{-64x^6y^9}$$. 2. **Recall the cube root properties:** - The cube root of a product is the product of the cube roots: $$\sqrt[3]{abc} = \sqrt[3]{a} \cdot \sqrt[3]{b} \cdot \sqrt[3]{c}$$. - For negative numbers, $$\sqrt[3]{-a} = -\sqrt[3]{a}$$. - For powers, $$\sqrt[3]{x^n} = x^{\frac{n}{3}}$$. 3. **Apply the cube root to each factor:** $$\sqrt[3]{-64x^6y^9} = \sqrt[3]{-64} \cdot \sqrt[3]{x^6} \cdot \sqrt[3]{y^9}$$ 4. **Simplify each cube root:** - $$\sqrt[3]{-64} = -4$$ because $$(-4)^3 = -64$$. - $$\sqrt[3]{x^6} = x^{\frac{6}{3}} = x^2$$. - $$\sqrt[3]{y^9} = y^{\frac{9}{3}} = y^3$$. 5. **Combine the simplified parts:** $$-4 \cdot x^2 \cdot y^3 = -4x^2y^3$$ **Final answer:** $$\boxed{-4x^2y^3}$$