Simplify 1. \( \frac{2^{x+2}-2^{x+1}}{2^{x}+2^{x+2}} \) \( 6=\frac{3^{x+1}+3^{x+2}}{8 \cdot 3^{x+1}} \) 3. \( \frac{4^{x}+3 \cdot 2^{2 x+1}}{7 \cdot 2^{2 x+1}} \) 4. \( \frac{12^{x}+4^{x} \cdot 3^{x+1}}{2^{2 x+4} \cdot 3^{x}} \) 5. \( \frac{2 \cdot 3^{x}+3^{x-2}}{3^{x+1}-7 \cdot 3^{x-1}} \)

Simplify the expression by following steps: - step0: Solution: \(\frac{\left(2\times 3^{x}+3^{x-2}\right)}{\left(3^{x+1}-7\times 3^{x-1}\right)}\) - step1: Remove the parentheses: \(\frac{2\times 3^{x}+3^{x-2}}{3^{x+1}-7\times 3^{x-1}}\) - step2: Subtract the terms: \(\frac{2\times 3^{x}+3^{x-2}}{2\times 3^{x-1}}\) - step3: Factor the expression: \(\frac{\left(2+3^{-2}\right)\times 3^{x}}{2\times 3^{x-1}}\) - step4: Reduce the fraction: \(\frac{\left(2+3^{-2}\right)\times 3}{2}\) - step5: Calculate: \(\frac{\frac{19}{3}}{2}\) - step6: Multiply by the reciprocal: \(\frac{19}{3}\times \frac{1}{2}\) - step7: Multiply the terms: \(\frac{19}{3\times 2}\) - step8: Multiply the terms: \(\frac{19}{6}\) Calculate or simplify the expression \( (2^(x+2)-2^(x+1))/(2^x+2^(x+2)) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(2^{x+2}-2^{x+1}\right)}{\left(2^{x}+2^{x+2}\right)}\) - step1: Remove the parentheses: \(\frac{2^{x+2}-2^{x+1}}{2^{x}+2^{x+2}}\) - step2: Subtract the terms: \(\frac{2^{x+1}}{2^{x}+2^{x+2}}\) - step3: Add the terms: \(\frac{2^{x+1}}{5\times 2^{x}}\) - step4: Divide the terms: \(\frac{2}{5}\) Calculate or simplify the expression \( (4^x+3*2^(2*x+1))/(7*2^(2*x+1)) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(4^{x}+3\times 2^{2x+1}\right)}{\left(7\times 2^{2x+1}\right)}\) - step1: Remove the parentheses: \(\frac{4^{x}+3\times 2^{2x+1}}{7\times 2^{2x+1}}\) - step2: Add the terms: \(\frac{7\times 2^{2x}}{7\times 2^{2x+1}}\) - step3: Reduce the fraction: \(\frac{2^{2x}}{2^{2x+1}}\) - step4: Divide the numbers: \(\frac{1}{2^{2x+1-2x}}\) - step5: Subtract the terms: \(\frac{1}{2^{1}}\) - step6: Simplify: \(\frac{1}{2}\) Calculate or simplify the expression \( (12^x+4^x*3^(x+1))/(2^(2*x+4)*3^x) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(12^{x}+4^{x}\times 3^{x+1}\right)}{\left(2^{2x+4}\times 3^{x}\right)}\) - step1: Remove the parentheses: \(\frac{12^{x}+4^{x}\times 3^{x+1}}{2^{2x+4}\times 3^{x}}\) - step2: Add the terms: \(\frac{4\times 12^{x}}{2^{2x+4}\times 3^{x}}\) - step3: Factor the expression: \(\frac{2^{2}\times 12^{x}}{2^{4+2x}\times 3^{x}}\) - step4: Reduce the fraction: \(\frac{12^{x}}{2^{2+2x}\times 3^{x}}\) - step5: Factor the expression: \(\frac{2^{2x}\times 3^{x}}{2^{2x+2}\times 3^{x}}\) - step6: Reduce the fraction: \(\frac{2^{2x}}{2^{2x+2}}\) - step7: Divide the numbers: \(\frac{1}{2^{2x+2-2x}}\) - step8: Subtract the terms: \(\frac{1}{2^{2}}\) - step9: Evaluate the power: \(\frac{1}{4}\) Let's simplify each expression step by step. 1. **Expression:** \[ \frac{2^{x+2}-2^{x+1}}{2^{x}+2^{x+2}} \] **Simplification:** - The numerator can be factored as \(2^{x+1}(2 - 1) = 2^{x+1}\). - The denominator can be factored as \(2^{x}(1 + 2^{2}) = 2^{x}(1 + 4) = 5 \cdot 2^{x}\). - Thus, the expression simplifies to: \[ \frac{2^{x+1}}{5 \cdot 2^{x}} = \frac{2}{5} \] 2. **Expression:** \[ 6=\frac{3^{x+1}+3^{x+2}}{8 \cdot 3^{x+1}} \] **Note:** This is an equation, so we need to solve it instead of simplifying. Let's proceed with solving it. 3. **Expression:** \[ \frac{4^{x}+3 \cdot 2^{2 x+1}}{7 \cdot 2^{2 x+1}} \] **Simplification:** - Rewrite \(4^{x}\) as \(2^{2x}\), so the numerator becomes \(2^{2x} + 3 \cdot 2^{2x + 1} = 2^{2x} + 6 \cdot 2^{2x} = 7 \cdot 2^{2x}\). - The denominator is \(7 \cdot 2^{2x + 1}\). - Thus, the expression simplifies to: \[ \frac{7 \cdot 2^{2x}}{7 \cdot 2^{2x + 1}} = \frac{1}{2} \] 4. **Expression:** \[ \frac{12^{x}+4^{x} \cdot 3^{x+1}}{2^{2 x+4} \cdot 3^{x}} \] **Simplification:** - Rewrite \(12^{x}\) as \(3^{x} \cdot 2^{2x}\) and \(4^{x}\) as \(2^{2x}\). - The numerator becomes \(3^{x} \cdot 2^{2x} + 2^{2x} \cdot 3^{x+1} = 3^{x} \cdot 2^{2x} + 2^{2x} \cdot 3^{x} \cdot 3 = 4 \cdot 3^{x} \cdot 2^{2x}\). - The denominator is \(2^{2x + 4} \cdot 3^{x}\). - Thus, the expression simplifies to: \[ \frac{4 \cdot 3^{x} \cdot 2^{2x}}{2^{2x + 4} \cdot 3^{x}} = \frac{4}{16} = \frac{1}{4} \] 5. **Expression:** \[ \frac{2 \cdot 3^{x}+3^{x-2}}{3^{x+1}-7 \cdot 3^{x-1}} \] **Simplification:** - The numerator can be factored as \(3^{x-2}(2 \cdot 3^{2} + 1) = 3^{x-2}(18 + 1) = 19 \cdot 3^{x-2}\). - The denominator can be factored as \(3^{x-1}(3 - 7) = -4 \cdot 3^{x-1}\). - Thus, the expression simplifies to: \[ \frac{19 \cdot 3^{x-2}}{-4 \cdot 3^{x-1}} = \frac{19}{-4 \cdot 3} = -\frac{19}{12} \] ### Summary of Results: 1. \( \frac{2}{5} \) 2. **Equation to solve:** \( 6=\frac{3^{x+1}+3^{x+2}}{8 \cdot 3^{x+1}} \) 3. \( \frac{1}{2} \) 4. \( \frac{1}{4} \) 5. \( -\frac{19}{12} \)