1. **State the problem:** Simplify the expression $$\sqrt[3]{56x^7y^5}$$ where $$x \neq 0$$ and $$y \neq 0$$. 2. **Recall the cube root property:** For any positive integers $$a, b$$ and variables, $$\sqrt[3]{a^3} = a$$ and $$\sqrt[3]{a^m} = a^{\frac{m}{3}}$$. 3. **Factor the radicand:** $$56x^7y^5 = 7 \times 8 \times x^7 \times y^5$$ 4. **Rewrite the cube root using factors:** $$\sqrt[3]{56x^7y^5} = \sqrt[3]{7 \times 8 \times x^7 \times y^5} = \sqrt[3]{7} \times \sqrt[3]{8} \times \sqrt[3]{x^7} \times \sqrt[3]{y^5}$$ 5. **Simplify cube roots of perfect cubes:** $$\sqrt[3]{8} = 2$$ because $$8 = 2^3$$. 6. **Simplify powers inside the cube root:** $$\sqrt[3]{x^7} = x^{\frac{7}{3}} = x^{2 + \frac{1}{3}} = x^2 \times x^{\frac{1}{3}} = x^2 \sqrt[3]{x}$$ $$\sqrt[3]{y^5} = y^{\frac{5}{3}} = y^{1 + \frac{2}{3}} = y \times y^{\frac{2}{3}} = y \sqrt[3]{y^2}$$ 7. **Combine all simplified parts:** $$2 \times x^2 \sqrt[3]{x} \times y \sqrt[3]{y^2} \times \sqrt[3]{7} = 2 x^2 y \sqrt[3]{7 x y^2}$$ 8. **Final answer:** $$\boxed{2 x^2 y \sqrt[3]{7 x y^2}}$$ which corresponds to option A.