1. \( \frac{2^{n} \sqrt{32}+2^{n} \sqrt{2}}{2^{n} \sqrt{50}} \)

1. Factor out \(2^n\) from the numerator: \[ \frac{2^n \sqrt{32} + 2^n \sqrt{2}}{2^n \sqrt{50}} = \frac{2^n (\sqrt{32} + \sqrt{2})}{2^n \sqrt{50}}. \] 2. Cancel the common term \(2^n\) in the numerator and denominator: \[ \frac{\sqrt{32} + \sqrt{2}}{\sqrt{50}}. \] 3. Simplify the square roots: \[ \sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2}, \quad \sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}. \] 4. Substitute the simplified forms into the expression: \[ \frac{4\sqrt{2} + \sqrt{2}}{5\sqrt{2}} = \frac{(4 + 1)\sqrt{2}}{5\sqrt{2}} = \frac{5\sqrt{2}}{5\sqrt{2}}. \] 5. Since the numerator and denominator are equal, the expression simplifies to: \[ 1. \]