Simplify completely: \( \frac{1+\frac{4}{x}}{1-\frac{16}{x^{2}}} \)

To simplify the expression \( \frac{1+\frac{4}{x}}{1-\frac{16}{x^{2}}} \), we first need to handle the fractions in both the numerator and the denominator. **Step 1: Simplifying the numerator** The numerator is: \[ 1 + \frac{4}{x} \] To combine this into a single fraction, we get: \[ \frac{x}{x} + \frac{4}{x} = \frac{x + 4}{x} \] **Step 2: Simplifying the denominator** The denominator is: \[ 1 - \frac{16}{x^2} \] We can rewrite this as: \[ \frac{x^2}{x^2} - \frac{16}{x^2} = \frac{x^2 - 16}{x^2} \] We recognize that \( x^2 - 16 \) is a difference of squares and can be factored: \[ x^2 - 16 = (x - 4)(x + 4) \] So the denominator becomes: \[ \frac{(x - 4)(x + 4)}{x^2} \] **Step 3: Putting it all together** Now we can substitute the simplified forms of the numerator and the denominator back into the original expression: \[ \frac{\frac{x + 4}{x}}{\frac{(x - 4)(x + 4)}{x^2}} = \frac{x + 4}{x} \cdot \frac{x^2}{(x - 4)(x + 4)} \] This leads to: \[ \frac{x(x + 4)}{(x - 4)(x + 4)} \] **Step 4: Canceling common factors** We can cancel \( x + 4 \) from the numerator and the denominator (assuming \( x \neq -4 \)): \[ \frac{x}{x - 4} \] **Final Result** Thus, the completely simplified expression is: \[ \boxed{\frac{x}{x-4}} \]