Add or subtract. Assume that all variables represent positive real numbers. \( \sqrt[3]{162 x y^{3}}-5 \sqrt[3]{48 x y^{3}}+y \sqrt[3]{48 x} \) \( \sqrt[3]{162 x y^{3}}-5 \sqrt[3]{48 x y^{3}}+y \sqrt[3]{48 x}=\square \) (Type an exact answer, using radicals as needed. Simplify your answer.)

Let's simplify the expression step by step. First, we can rewrite the terms involving cube roots: 1. The first term is \( \sqrt[3]{162xy^3} \). 2. The second term is \( -5\sqrt[3]{48xy^3} \). 3. The third term is \( y\sqrt[3]{48x} \). Next, we'll factor out the common cube root components where possible. Begin with the first two terms: We notice that \( 162 = 2 \times 81 = 2 \times 3^4 \) and \( 48 = 16 \times 3 = 4^2 \times 3 \). We can rewrite the terms: 1. \( \sqrt[3]{162xy^3} = \sqrt[3]{(2 \cdot 3^4) \cdot x \cdot (y^3)} = 3\sqrt[3]{18xy^3} \) 2. \( 5\sqrt[3]{48xy^3} = 5\sqrt[3]{(16 \cdot 3)xy^3} = 5\cdot 2\sqrt[3]{12xy^3} = 10\sqrt[3]{12xy^3} \) Now, we can combine the like terms using the expression for simplification: 1. The simplified expression \( \sqrt[3]{162xy^3} - 5\sqrt[3]{48xy^3} + y\sqrt[3]{48x} \) can be written as: \( \sqrt[3]{(162 - 240 + 48y)x(y^2)} = \sqrt[3]{\text{expression}} \) in unified terms. Calculating: Combining constants, we have: - \( 162 - 240 + 48y\) While we cannot derive a perfect factorization here across different components, we will simplify under a common radical where possible, ultimately deriving: Therefore, our combined and simplified expression is: \[ \sqrt[3]{18xy^3} - 10\sqrt[3]{12xy^3} + y\sqrt[3]{48x} = 3 \sqrt[3]{xy^3}. \] The final answer is: \[ \sqrt[3]{(12y + 162 - 240)xy^3} = \sqrt[3]{(12y - 78)xy^3}. \] Thus the answer is: \[ 3\sqrt[3]{xy^3} + y\sqrt[3]{48x}. \] So altogether, we can simplify this down to: \[ 3\sqrt[3]{xy^3} + y\sqrt[3]{48x} = \square. \] Here, however, you will notice that simplifications and algebraic structures may offer diverse perspectives on representation. Students frequently miss factoring altogether when troubleshooting their equation for radical expressions, so keep an eye on common factors!