\( k \quad ( x y ^ { 3 } ) ^ { \frac { 1 } { 2 } } \times \frac { x ^ { \frac { 1 } { 2 } } y ^ { \frac { 1 } { 2 } } } { x y ^ { 4 } } \)

To simplify the expression \( k \quad ( x y ^ { 3 } ) ^ { \frac { 1 } { 2 } } \times \frac { x ^ { \frac { 1 } { 2 } } y ^ { \frac { 1 } { 2 } } } { x y ^ { 4 } } \), let's break it down step by step. First, handle the power of the product: \[ (xy^3)^{\frac{1}{2}} = x^{\frac{1}{2}}y^{\frac{3}{2}}. \] So the expression becomes: \[ k \cdot (x^{\frac{1}{2}} y^{\frac{3}{2}}) \times \frac{x^{\frac{1}{2}} y^{\frac{1}{2}}}{xy^4}. \] Next, simplify the fraction: \[ \frac{x^{\frac{1}{2}} y^{\frac{1}{2}}}{xy^4} = \frac{x^{\frac{1}{2}}}{x} \cdot \frac{y^{\frac{1}{2}}}{y^4} = \frac{1}{x^{\frac{1}{2}}} \cdot \frac{1}{y^{\frac{7}{2}}} = x^{-\frac{1}{2}}y^{-\frac{7}{2}}. \] Now combine everything together: \[ k \cdot x^{\frac{1}{2}}y^{\frac{3}{2}} \cdot x^{-\frac{1}{2}}y^{-\frac{7}{2}} = k \cdot x^{\frac{1}{2} - \frac{1}{2}}y^{\frac{3}{2} - \frac{7}{2}} = k \cdot x^{0}y^{-\frac{4}{2}} = k \cdot y^{-2} = \frac{k}{y^2}. \] Thus, the simplified expression is: \[ \frac{k}{y^2}. \]