Question 1 \( 1 \cdot 1 \frac{9^{x}+3^{2 x+1}}{18^{x} \cdot 2^{-x}} \)

To simplify the expression: \[ \frac{9^{x} + 3^{2x + 1}}{18^{x} \cdot 2^{-x}} \] Let's break it down step by step. ### Step 1: Rewrite the Terms with Powers of 3 and 2 1. **Express \(9^{x}\) in terms of \(3\):** \[ 9^{x} = (3^2)^x = 3^{2x} \] 2. **Express \(3^{2x + 1}\) as a product:** \[ 3^{2x + 1} = 3^{2x} \cdot 3^1 = 3^{2x} \cdot 3 \] 3. **Express \(18^{x}\) in terms of \(2\) and \(3\):** \[ 18^{x} = (2 \cdot 9)^x = 2^{x} \cdot 9^{x} = 2^{x} \cdot 3^{2x} \] 4. **Express \(2^{-x}\) as a reciprocal:** \[ 2^{-x} = \frac{1}{2^{x}} \] ### Step 2: Substitute Back into the Original Expression \[ \frac{3^{2x} + 3^{2x} \cdot 3}{2^{x} \cdot 3^{2x} \cdot \frac{1}{2^{x}}} \] ### Step 3: Simplify the Denominator \[ 2^{x} \cdot 3^{2x} \cdot \frac{1}{2^{x}} = 3^{2x} \] ### Step 4: Simplify the Entire Expression \[ \frac{3^{2x} + 3^{2x} \cdot 3}{3^{2x}} = \frac{3^{2x}(1 + 3)}{3^{2x}} = 1 + 3 = 4 \] ### Final Answer \[ 4 \]