e) \( \frac{x^{x+2}-2^{x+3}}{2^{x+1}-2^{x+2}} \)

Simplify the expression by following steps: - step0: Solution: \(\frac{\left(x^{x+2}-2^{x+3}\right)}{\left(2^{x+1}-2^{x+2}\right)}\) - step1: Remove the parentheses: \(\frac{x^{x+2}-2^{x+3}}{2^{x+1}-2^{x+2}}\) - step2: Subtract the terms: \(\frac{x^{x+2}-2^{x+3}}{-2^{x+1}}\) - step3: Rewrite the fraction: \(-\frac{x^{x+2}-2^{x+3}}{2^{x+1}}\) To simplify the expression \[ \frac{x^{x+2}-2^{x+3}}{2^{x+1}-2^{x+2}}, \] we can follow these steps: 1. **Simplify the denominator**: The denominator can be factored as follows: \[ 2^{x+1} - 2^{x+2} = 2^{x+1}(1 - 2) = -2^{x+1}. \] 2. **Rewrite the expression**: Substituting the simplified denominator back into the expression gives us: \[ \frac{x^{x+2} - 2^{x+3}}{-2^{x+1}}. \] 3. **Final expression**: This can be rewritten as: \[ -\frac{x^{x+2} - 2^{x+3}}{2^{x+1}}. \] Thus, the simplified form of the original expression is \[ -\frac{x^{x+2} - 2^{x+3}}{2^{x+1}}. \]