Simplify. \[ \frac{\frac{1}{x}+\frac{5}{y}}{\frac{7}{x}+\frac{1}{y^{2}}} \]

To simplify the expression \[ \frac{\frac{1}{x}+\frac{5}{y}}{\frac{7}{x}+\frac{1}{y^{2}}}, \] we start by finding a common denominator for both the numerator and the denominator. **Step 1: Simplifying the Numerator** The numerator is \[ \frac{1}{x} + \frac{5}{y}. \] The common denominator for \(x\) and \(y\) is \(xy\). Rewriting each term with this common denominator: \[ \frac{1}{x} = \frac{y}{xy}, \quad \frac{5}{y} = \frac{5x}{xy}. \] Now, we combine these fractions: \[ \frac{1}{x} + \frac{5}{y} = \frac{y + 5x}{xy}. \] **Step 2: Simplifying the Denominator** Next, we simplify the denominator: \[ \frac{7}{x} + \frac{1}{y^{2}}. \] The common denominator for \(x\) and \(y^{2}\) is \(xy^{2}\). Rewriting each term with the common denominator: \[ \frac{7}{x} = \frac{7y^{2}}{xy^{2}}, \quad \frac{1}{y^{2}} = \frac{x}{xy^{2}}. \] Now we combine these: \[ \frac{7}{x} + \frac{1}{y^{2}} = \frac{7y^{2} + x}{xy^{2}}. \] **Step 3: Putting it Together** Now we can rewrite the entire expression: \[ \frac{\frac{y + 5x}{xy}}{\frac{7y^{2} + x}{xy^{2}}}. \] To divide by a fraction, we multiply by its reciprocal: \[ \frac{y + 5x}{xy} \cdot \frac{xy^{2}}{7y^{2} + x}. \] The \(xy\) in the numerator and denominator will cancel out: \[ \frac{(y + 5x) y^{2}}{7y^{2} + x}. \] Thus, the expression simplifies to: \[ \frac{(y + 5x) y^{2}}{7y^{2} + x}. \] In conclusion, the simplified form of the original expression is: \[ \boxed{\frac{(y + 5x) y^{2}}{7y^{2} + x}}. \]