2. Simplify: \( \begin{array}{ll}\text { a) } \frac{2^{n+1}+2^{n+3}}{2^{n+1}} & \text { b) } \frac{3^{x+2}-3^{x}}{3^{x+1}+3^{x}}\end{array} \)

Let's simplify each expression step by step. ### a) \( \frac{2^{n+1} + 2^{n+3}}{2^{n+1}} \) 1. **Factor out \( 2^{n+1} \) in the numerator:** \[ \frac{2^{n+1} (1 + 2^{2})}{2^{n+1}} \] 2. **Simplify \( 2^2 \):** \[ \frac{2^{n+1} (1 + 4)}{2^{n+1}} = \frac{2^{n+1} \times 5}{2^{n+1}} \] 3. **Cancel out \( 2^{n+1} \):** \[ 5 \] **Simplified Result:** \( 5 \) --- ### b) \( \frac{3^{x+2} - 3^{x}}{3^{x+1} + 3^{x}} \) 1. **Factor out \( 3^{x} \) in the numerator and denominator:** \[ \frac{3^{x} (3^{2} - 1)}{3^{x} (3^{1} + 1)} \] 2. **Simplify the exponents:** \[ \frac{3^{x} (9 - 1)}{3^{x} (3 + 1)} = \frac{3^{x} \times 8}{3^{x} \times 4} \] 3. **Cancel out \( 3^{x} \):** \[ \frac{8}{4} = 2 \] **Simplified Result:** \( 2 \) --- **Final Answers:** - **a)** \( 5 \) - **b)** \( 2 \)