1. **Problem:** Simplify the expression $$11\sqrt[3]{x^4 y^7} - 5xy \sqrt[3]{8xy^4} + y^2 \sqrt[3]{27x^4 y}$$ and verify it equals $$4xy^2 \sqrt[3]{xy}$$. 2. **Formula and rules:** Recall that $$\sqrt[3]{a^m b^n} = a^{\frac{m}{3}} b^{\frac{n}{3}}$$ and that $$a^{p} a^{q} = a^{p+q}$$. Also, $$\sqrt[3]{a^3} = a$$. 3. **Step-by-step simplification:** - First term: $$11 \sqrt[3]{x^4 y^7} = 11 x^{\frac{4}{3}} y^{\frac{7}{3}} = 11 x^{1 + \frac{1}{3}} y^{2 + \frac{1}{3}} = 11 x y^2 x^{\frac{1}{3}} y^{\frac{1}{3}} = 11 x y^2 \sqrt[3]{x y}$$. - Second term: $$-5 x y \sqrt[3]{8 x y^4} = -5 x y \sqrt[3]{2^3 x y^4} = -5 x y \cdot 2 \sqrt[3]{x y^4} = -10 x y \cdot x^{\frac{1}{3}} y^{\frac{4}{3}} = -10 x y \cdot x^{\frac{1}{3}} y^{1 + \frac{1}{3}} = -10 x y^{2} x^{\frac{1}{3}} y^{\frac{1}{3}} = -10 x y^{2} \sqrt[3]{x y}$$. - Third term: $$y^{2} \sqrt[3]{27 x^{4} y} = y^{2} \sqrt[3]{3^{3} x^{4} y} = y^{2} \cdot 3 \sqrt[3]{x^{4} y} = 3 y^{2} x^{\frac{4}{3}} y^{\frac{1}{3}} = 3 y^{2} x^{1 + \frac{1}{3}} y^{\frac{1}{3}} = 3 x y^{2} x^{\frac{1}{3}} y^{\frac{1}{3}} = 3 x y^{2} \sqrt[3]{x y}$$. 4. **Combine all terms:** $$11 x y^{2} \sqrt[3]{x y} - 10 x y^{2} \sqrt[3]{x y} + 3 x y^{2} \sqrt[3]{x y} = (11 - 10 + 3) x y^{2} \sqrt[3]{x y} = 4 x y^{2} \sqrt[3]{x y}$$. 5. **Final answer:** $$\boxed{4 x y^{2} \sqrt[3]{x y}}$$. This matches the given answer, confirming the simplification is correct.