b) 2) \( \frac{4^{x}+3.2^{2 x+1}}{7.2^{2 x+1}} \)

Simplify the expression by following steps: - step0: Solution: \(\frac{4^{x}+3\times 2^{2x+1}}{7\times 2^{2x+1}}\) - step1: Add the terms: \(\frac{7\times 2^{2x}}{7\times 2^{2x+1}}\) - step2: Reduce the fraction: \(\frac{2^{2x}}{2^{2x+1}}\) - step3: Divide the numbers: \(\frac{1}{2^{2x+1-2x}}\) - step4: Subtract the terms: \(\frac{1}{2^{1}}\) - step5: Simplify: \(\frac{1}{2}\) To simplify the expression \( \frac{4^{x}+3 \cdot 2^{2 x+1}}{7 \cdot 2^{2 x+1}} \), we can follow these steps: 1. **Rewrite \(4^x\)**: We know that \(4^x\) can be expressed as \((2^2)^x = 2^{2x}\). Thus, we can rewrite the expression as: \[ \frac{2^{2x} + 3 \cdot 2^{2x + 1}}{7 \cdot 2^{2x + 1}} \] 2. **Factor out \(2^{2x}\)** from the numerator: \[ = \frac{2^{2x}(1 + 3 \cdot 2)}{7 \cdot 2^{2x + 1}} \] Here, \(3 \cdot 2 = 6\), so we have: \[ = \frac{2^{2x}(1 + 6)}{7 \cdot 2^{2x + 1}} = \frac{2^{2x} \cdot 7}{7 \cdot 2^{2x + 1}} \] 3. **Cancel out \(7\)** from the numerator and denominator: \[ = \frac{2^{2x}}{2^{2x + 1}} \] 4. **Simplify the powers of \(2\)**: \[ = \frac{1}{2} \] Thus, the simplified form of the expression is: \[ \frac{1}{2} \]