What is the simplified form of the following expression? Assume \( x \geq 0 \) and \( y \geq 0 \). \( 2(\sqrt[4]{16 x})-2(\sqrt[4]{2 y})+3(\sqrt[4]{81 x})-4(\sqrt[4]{32 y}) \)

To simplify the expression \( 2(\sqrt[4]{16 x})-2(\sqrt[4]{2 y})+3(\sqrt[4]{81 x})-4(\sqrt[4]{32 y}) \), we start by simplifying each individual term. 1. **Calculate \( \sqrt[4]{16x} \)**: \[ \sqrt[4]{16x} = \sqrt[4]{16} \cdot \sqrt[4]{x} = 2 \cdot \sqrt[4]{x} \] Thus, \( 2(\sqrt[4]{16 x}) = 2(2 \sqrt[4]{x}) = 4 \sqrt[4]{x} \). 2. **Calculate \( \sqrt[4]{2y} \)**: \[ \sqrt[4]{2y} = \sqrt[4]{2} \cdot \sqrt[4]{y} \] Therefore, \( -2(\sqrt[4]{2 y}) = -2(\sqrt[4]{2} \sqrt[4]{y}) = -2\sqrt[4]{2} \sqrt[4]{y} \). 3. **Calculate \( \sqrt[4]{81x} \)**: \[ \sqrt[4]{81x} = \sqrt[4]{81} \cdot \sqrt[4]{x} = 3 \cdot \sqrt[4]{x} \] Thus, \( 3(\sqrt[4]{81 x}) = 3(3 \sqrt[4]{x}) = 9 \sqrt[4]{x} \). 4. **Calculate \( \sqrt[4]{32y} \)**: \[ \sqrt[4]{32y} = \sqrt[4]{32} \cdot \sqrt[4]{y} = 2\sqrt[4]{2} \cdot \sqrt[4]{y} \] Therefore, \( -4(\sqrt[4]{32 y}) = -4(2\sqrt[4]{2} \sqrt[4]{y}) = -8\sqrt[4]{2} \sqrt[4]{y} \). Now, combining all the terms: \[ 4 \sqrt[4]{x} + 9 \sqrt[4]{x} - 2\sqrt[4]{2} \sqrt[4]{y} - 8\sqrt[4]{2} \sqrt[4]{y} \] Combine the like terms: \[ (4 + 9) \sqrt[4]{x} + (-2\sqrt[4]{2} - 8\sqrt[4]{2}) \sqrt[4]{y} \] This gives us: \[ 13 \sqrt[4]{x} - 10\sqrt[4]{2} \sqrt[4]{y} \] Thus, the simplified form of the expression \( 2(\sqrt[4]{16 x})-2(\sqrt[4]{2 y})+3(\sqrt[4]{81 x})-4(\sqrt[4]{32 y}) \) is: \[ \boxed{13 \sqrt[4]{x} - 10\sqrt[4]{2} \sqrt[4]{y}} \]