1. **State the problem:** Simplify the expression $$\sqrt[3]{512x^4 y^5}$$. 2. **Recall the cube root properties:** The cube root of a product is the product of the cube roots: $$\sqrt[3]{abc} = \sqrt[3]{a} \cdot \sqrt[3]{b} \cdot \sqrt[3]{c}$$. 3. **Break down the expression:** $$\sqrt[3]{512x^4 y^5} = \sqrt[3]{512} \cdot \sqrt[3]{x^4} \cdot \sqrt[3]{y^5}$$ 4. **Simplify each part:** - $$512 = 8^3$$, so $$\sqrt[3]{512} = 8$$. - For $$x^4$$, write as $$x^{3+1} = x^3 \cdot x^1$$, so $$\sqrt[3]{x^4} = \sqrt[3]{x^3 \cdot x} = \sqrt[3]{x^3} \cdot \sqrt[3]{x} = x \cdot \sqrt[3]{x}$$. - For $$y^5$$, write as $$y^{3+2} = y^3 \cdot y^2$$, so $$\sqrt[3]{y^5} = \sqrt[3]{y^3 \cdot y^2} = \sqrt[3]{y^3} \cdot \sqrt[3]{y^2} = y \cdot \sqrt[3]{y^2}$$. 5. **Combine all simplified parts:** $$8 \cdot x \cdot y \cdot \sqrt[3]{x} \cdot \sqrt[3]{y^2} = 8xy \sqrt[3]{xy^2}$$. **Final answer:** $$8xy \sqrt[3]{xy^2}$$