1. **State the problem:** Simplify the expression $$(125 \cdot 7)^{\frac{1}{3}} \cdot x^{\frac{5}{3}} \cdot y^{\frac{9}{3}}$$. 2. **Rewrite the expression using properties of exponents:** $$ (125)^{\frac{1}{3}} \cdot (7)^{\frac{1}{3}} \cdot x^{\frac{5}{3}} \cdot y^{3} $$ 3. **Simplify the cube root of 125:** Since $125 = 5^3$, then $$ (5^3)^{\frac{1}{3}} = 5^{3 \times \frac{1}{3}} = 5^1 = 5 $$ 4. **Rewrite the expression with simplified terms:** $$ 5 \cdot 7^{\frac{1}{3}} \cdot x^{\frac{5}{3}} \cdot y^{3} $$ 5. **Split the exponent of $x$ into sum of fractions:** $$ x^{\frac{5}{3}} = x^{1 + \frac{2}{3}} = x^1 \cdot x^{\frac{2}{3}} $$ 6. **Rewrite the expression:** $$ 5 \cdot 7^{\frac{1}{3}} \cdot x \cdot x^{\frac{2}{3}} \cdot y^{3} $$ 7. **Group terms to express as a cube root:** $$ 5 x y^3 \cdot 7^{\frac{1}{3}} \cdot x^{\frac{2}{3}} = 5 x y^3 \cdot \sqrt[3]{7 x^2} $$ 8. **Combine all under a single cube root:** $$ \sqrt[3]{5^3} \cdot \sqrt[3]{7} \cdot \sqrt[3]{x^2} \cdot y^3 = \sqrt[3]{5^3 \cdot 7 \cdot x^2 \cdot y^9} = \sqrt[3]{875 x^5 y^9} $$ **Final simplified form:** $$ 5 x y^3 \sqrt[3]{7 x^2} = \sqrt[3]{875 x^5 y^9} $$