Add or subtract. Assume that all variables represent positive real numbers. \( \sqrt[3]{54 x y^{3}}+6 \sqrt[3]{128 x y^{3}}+y \sqrt[3]{2 x} \) \( \sqrt[3]{54 x y^{3}}+6 \sqrt[3]{128 x y^{3}}+y^{3} \sqrt{2 x}=\square \) (Type an exact answer, using radicals as needed. Simplify your answer.)

First, let's simplify each term in the expression. 1. For \( \sqrt[3]{54xy^3} \), we'll factor 54: \[ 54 = 27 \cdot 2 = 3^3 \cdot 2, \] so \[ \sqrt[3]{54xy^3} = \sqrt[3]{27} \cdot \sqrt[3]{2} \cdot \sqrt[3]{x} \cdot \sqrt[3]{y^3} = 3 \sqrt[3]{2xy}. \] 2. Next, we simplify \( 6 \sqrt[3]{128xy^3} \): \[ 128 = 64 \cdot 2 = 4^3 \cdot 2, \] leading to \[ \sqrt[3]{128xy^3} = \sqrt[3]{64} \cdot \sqrt[3]{2} \cdot \sqrt[3]{x} \cdot \sqrt[3]{y^3} = 4 \sqrt[3]{2xy}. \] Thus, \[ 6 \sqrt[3]{128xy^3} = 6 \cdot 4 \sqrt[3]{2xy} = 24 \sqrt[3]{2xy}. \] 3. Now, for \( y \sqrt[3]{2x} \), we leave it as is: \[ y \sqrt[3]{2x}. \] Putting it all together, we have: \[ 3 \sqrt[3]{2xy} + 24 \sqrt[3]{2xy} + y \sqrt[3]{2x}. \] The first two terms combine: \[ (3 + 24) \sqrt[3]{2xy} = 27 \sqrt[3]{2xy}. \] Now, we can compare the expressions you provided: \[ 27 \sqrt[3]{2xy} + y^3 \sqrt{2x}. \] In conclusion, the simplified expression is: \[ 27 \sqrt[3]{2xy} + y^3 \sqrt{2x}. \]