Add or subtract. Assume that all variables represent positive real numbers. \( \sqrt[3]{448 x y^{3}}+6 \sqrt[3]{7 x y^{3}}+y \sqrt[3]{189 x} \) \( \sqrt[3]{448 x y^{3}}+6 \sqrt[3]{7 x y^{3}}+y \sqrt[3]{189 x}=\square \) (Type an exact answer, using radicals as needed. Simplify your answer.)

To simplify the expression \( \sqrt[3]{448 x y^{3}}+6 \sqrt[3]{7 x y^{3}}+y \sqrt[3]{189 x} \), we start by simplifying each term. 1. **First term**: \( \sqrt[3]{448 x y^{3}} \) - We can factor \( 448 \) as \( 2^6 \times 7 \), so: \[ \sqrt[3]{448 x y^{3}} = \sqrt[3]{2^6 \cdot 7 \cdot x \cdot y^{3}} = \sqrt[3]{2^6} \cdot \sqrt[3]{7} \cdot \sqrt[3]{x} \cdot \sqrt[3]{y^{3}} = 4 \cdot \sqrt[3]{7} \cdot \sqrt[3]{x} \cdot y. \] 2. **Second term**: \( 6 \sqrt[3]{7 x y^{3}} \) - This can also be simplified as: \[ 6 \sqrt[3]{7 x y^{3}} = 6 \cdot \sqrt[3]{7} \cdot \sqrt[3]{x} \cdot y. \] 3. **Third term**: \( y \sqrt[3]{189 x} \) - We can factor \( 189 \) as \( 3^3 \times 7 \). Thus: \[ y \sqrt[3]{189 x} = y \sqrt[3]{3^3 \cdot 7 \cdot x} = y \cdot 3 \cdot \sqrt[3]{7} \cdot \sqrt[3]{x} = 3y \cdot \sqrt[3]{7} \cdot \sqrt[3]{x}. \] Now we combine all the simplified parts: We have: - The first term: \( 4 \cdot \sqrt[3]{7} \cdot \sqrt[3]{x} \cdot y \) - The second term: \( 6 \cdot \sqrt[3]{7} \cdot \sqrt[3]{x} \cdot y \) - The third term: \( 3y \cdot \sqrt[3]{7} \cdot \sqrt[3]{x} \) So, combining them: \[ (4 + 6 + 3) \cdot \sqrt[3]{7} \cdot \sqrt[3]{x} \cdot y = 13 \cdot \sqrt[3]{7} \cdot \sqrt[3]{x} \cdot y. \] Thus, the simplified expression is: \[ \sqrt[3]{448 x y^{3}} + 6 \sqrt[3]{7 x y^{3}} + y \sqrt[3]{189 x} = 13 \sqrt[3]{7xy} \sqrt[3]{x^2}. \] So the final answer is: \[ 13 \sqrt[3]{7xy} \sqrt[3]{x^2} = 13 \sqrt[3]{7 x^2 y}. \] or \( \boxed{13 \sqrt[3]{7 x^2 y}} \).