Simplify the following: \( \begin{array}{lll}\text { (a) } \frac{10^{x} \cdot 25^{x+1}}{5^{x} \cdot 50^{x-1}} & \text { (b) } \frac{6^{n+2} \times 10^{n-2}}{4^{n} \times 15^{n-2}} & \text { (c) } \frac{6^{x} \cdot 9^{x+1} \cdot 2}{27^{x+1} \cdot 2^{x-1}} \\ \text { (d) } \frac{8^{n} \cdot 6^{n-3} \cdot 9^{1-n}}{16^{n-1} \cdot 3^{-n}} & \text { (e) } \frac{2^{x+2}-2^{x+3}}{2^{x+1}-2^{x+2}} & \text { (f) } \frac{9^{x}+3^{x+1}}{18^{x} \cdot 2^{1-x}} \\ \text { (g) } \frac{3 \cdot 2^{x}-2^{x-1}}{2^{x}+2^{x+2}} & \text { (h) } \frac{4^{x}+2^{2 x-1}}{2^{2 x-1}} & \text { (i) } \frac{2.3^{x+2}+3^{x-3}}{5 \cdot 3^{x-2}}\end{array} \)

Simplify the expression by following steps: - step0: Solution: \(\frac{\left(4^{x}+2^{2x-1}\right)}{2^{2x-1}}\) - step1: Remove the parentheses: \(\frac{4^{x}+2^{2x-1}}{2^{2x-1}}\) - step2: Add the terms: \(\frac{3\times 2^{2x-1}}{2^{2x-1}}\) - step3: Rewrite the expression: \(3\times 1\) - step4: Multiply: \(3\) Calculate or simplify the expression \( (10^x * 25^(x+1)) / (5^x * 50^(x-1)) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(10^{x}\times 25^{x+1}\right)}{\left(5^{x}\times 50^{x-1}\right)}\) - step1: Remove the parentheses: \(\frac{10^{x}\times 25^{x+1}}{5^{x}\times 50^{x-1}}\) - step2: Factor the expression: \(\frac{5^{x}\times 2^{x}\times 25^{x+1}}{5^{x}\times 50^{x-1}}\) - step3: Reduce the fraction: \(\frac{2^{x}\times 25^{x+1}}{50^{x-1}}\) - step4: Factor the expression: \(\frac{2^{x}\times 25^{x+1}}{2^{x-1}\times 25^{x-1}}\) - step5: Reduce the fraction: \(2\times 25^{2}\) - step6: Evaluate the power: \(2\times 625\) - step7: Multiply: \(1250\) Calculate or simplify the expression \( (2 * 3^(x+2) + 3^(x-3)) / (5 * 3^(x-2)) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(2\times 3^{x+2}+3^{x-3}\right)}{\left(5\times 3^{x-2}\right)}\) - step1: Remove the parentheses: \(\frac{2\times 3^{x+2}+3^{x-3}}{5\times 3^{x-2}}\) - step2: Add the terms: \(\frac{487\times 3^{x-3}}{5\times 3^{x-2}}\) - step3: Reduce the fraction: \(\frac{487}{5\times 3}\) - step4: Calculate: \(\frac{487}{15}\) Calculate or simplify the expression \( (2^(x+2) - 2^(x+3)) / (2^(x+1) - 2^(x+2)) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(2^{x+2}-2^{x+3}\right)}{\left(2^{x+1}-2^{x+2}\right)}\) - step1: Remove the parentheses: \(\frac{2^{x+2}-2^{x+3}}{2^{x+1}-2^{x+2}}\) - step2: Subtract the terms: \(\frac{-2^{x+2}}{2^{x+1}-2^{x+2}}\) - step3: Subtract the terms: \(\frac{-2^{x+2}}{-2^{x+1}}\) - step4: Calculate: \(2^{x+2-\left(x+1\right)}\) - step5: Calculate: \(2\) Calculate or simplify the expression \( (6^x * 9^(x+1) * 2) / (27^(x+1) * 2^(x-1)) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(6^{x}\times 9^{x+1}\times 2\right)}{\left(27^{x+1}\times 2^{x-1}\right)}\) - step1: Remove the parentheses: \(\frac{6^{x}\times 9^{x+1}\times 2}{27^{x+1}\times 2^{x-1}}\) - step2: Multiply by \(a^{-n}:\) \(\frac{6^{x}\times 9^{x+1}\times 2\times 2^{-\left(x-1\right)}}{27^{x+1}}\) - step3: Calculate: \(\frac{6^{x}\times 9^{x+1}\times 2\times 2^{-x+1}}{27^{x+1}}\) - step4: Multiply: \(\frac{6^{x}\times 9^{x+1}\times 2^{2-x}}{27^{x+1}}\) - step5: Factor the expression: \(\frac{3^{x}\times 2^{x}\times 9^{x+1}\times 2^{2-x}}{3^{3x+3}}\) - step6: Reduce the fraction: \(\frac{2^{x}\times 9^{x+1}\times 2^{2-x}}{3^{2x+3}}\) - step7: Factor the expression: \(\frac{2^{x}\times 3^{2x+2}\times 2^{2-x}}{3^{2x+3}}\) - step8: Reduce the fraction: \(\frac{2^{x}\times 2^{2-x}}{3}\) - step9: Multiply the terms: \(\frac{2^{2}}{3}\) - step10: Evaluate the power: \(\frac{4}{3}\) Calculate or simplify the expression \( (6^(n+2) * 10^(n-2)) / (4^n * 15^(n-2)) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(6^{n+2}\times 10^{n-2}\right)}{\left(4^{n}\times 15^{n-2}\right)}\) - step1: Remove the parentheses: \(\frac{6^{n+2}\times 10^{n-2}}{4^{n}\times 15^{n-2}}\) - step2: Factor the expression: \(\frac{2^{n+2}\times 3^{n+2}\times 10^{n-2}}{2^{2n}\times 15^{n-2}}\) - step3: Reduce the fraction: \(\frac{3^{n+2}\times 10^{n-2}}{2^{n-2}\times 15^{n-2}}\) - step4: Factor the expression: \(\frac{3^{n+2}\times 10^{n-2}}{2^{n-2}\times 3^{n-2}\times 5^{n-2}}\) - step5: Reduce the fraction: \(\frac{3^{4}\times 10^{n-2}}{2^{n-2}\times 5^{n-2}}\) - step6: Factor the expression: \(\frac{3^{4}\times 2^{n-2}\times 5^{n-2}}{2^{n-2}\times 5^{n-2}}\) - step7: Reduce the fraction: \(3^{4}\) - step8: Evaluate the power: \(81\) Calculate or simplify the expression \( (3 * 2^x - 2^(x-1)) / (2^x + 2^(x+2)) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(3\times 2^{x}-2^{x-1}\right)}{\left(2^{x}+2^{x+2}\right)}\) - step1: Remove the parentheses: \(\frac{3\times 2^{x}-2^{x-1}}{2^{x}+2^{x+2}}\) - step2: Add the terms: \(\frac{3\times 2^{x}-2^{x-1}}{5\times 2^{x}}\) - step3: Factor the expression: \(\frac{\left(3-2^{-1}\right)\times 2^{x}}{5\times 2^{x}}\) - step4: Reduce the fraction: \(\frac{3-2^{-1}}{5}\) - step5: Calculate: \(\frac{\frac{5}{2}}{5}\) - step6: Multiply by the reciprocal: \(\frac{5}{2}\times \frac{1}{5}\) - step7: Reduce the numbers: \(\frac{1}{2}\times 1\) - step8: Multiply: \(\frac{1}{2}\) Calculate or simplify the expression \( (9^x + 3^(x+1)) / (18^x * 2^(1-x)) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(9^{x}+3^{x+1}\right)}{\left(18^{x}\times 2^{1-x}\right)}\) - step1: Remove the parentheses: \(\frac{9^{x}+3^{x+1}}{18^{x}\times 2^{1-x}}\) - step2: Multiply the terms: \(\frac{9^{x}+3^{x+1}}{2\times 9^{x}}\) - step3: Factor the expression: \(\frac{3^{x}\left(3^{x}+3\right)}{2\times 3^{2x}}\) - step4: Reduce the fraction: \(\frac{3^{x}+3}{2\times 3^{x}}\) - step5: Factor the expression: \(\frac{3\left(3^{x-1}+1\right)}{2\times 3^{x}}\) - step6: Divide the numbers: \(\frac{3^{1-x}\left(3^{x-1}+1\right)}{2}\) - step7: Calculate: \(\frac{1+3^{1-x}}{2}\) Calculate or simplify the expression \( (8^n * 6^(n-3) * 9^(1-n)) / (16^(n-1) * 3^(-n)) \). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(8^{n}\times 6^{n-3}\times 9^{1-n}\right)}{\left(16^{n-1}\times 3^{-n}\right)}\) - step1: Remove the parentheses: \(\frac{8^{n}\times 6^{n-3}\times 9^{1-n}}{16^{n-1}\times 3^{-n}}\) - step2: Factor the expression: \(\frac{2^{3n}\times 6^{n-3}\times 9^{1-n}}{2^{4n-4}\times 3^{-n}}\) - step3: Reduce the fraction: \(\frac{6^{n-3}\times 9^{1-n}}{2^{n-4}\times 3^{-n}}\) - step4: Factor the expression: \(\frac{2^{n-3}\times 3^{n-3}\times 9^{1-n}}{2^{n-4}\times 3^{-n}}\) - step5: Reduce the fraction: \(2\times 3^{2n-3}\times 9^{1-n}\) - step6: Transform the expression: \(2\times 3^{2n-3}\times 3^{2\left(1-n\right)}\) - step7: Multiply the terms: \(2\times 3^{2n-3+2\left(1-n\right)}\) - step8: Calculate: \(2\times 3^{-1}\) - step9: Use the properties of exponents: \(2\times \frac{1}{3}\) - step10: Multiply: \(\frac{2}{3}\) Here are the simplified results for each expression: (a) \[ \frac{10^{x} \cdot 25^{x+1}}{5^{x} \cdot 50^{x-1}} = 1250 \] (b) \[ \frac{6^{n+2} \times 10^{n-2}}{4^{n} \times 15^{n-2}} = 81 \] (c) \[ \frac{6^{x} \cdot 9^{x+1} \cdot 2}{27^{x+1} \cdot 2^{x-1}} = \frac{4}{3} \quad \text{(or } 1\frac{1}{3} \text{ or } 1.\dot{3} \text{)} \] (d) \[ \frac{8^{n} \cdot 6^{n-3} \cdot 9^{1-n}}{16^{n-1} \cdot 3^{-n}} = \frac{2}{3} \quad \text{(or } 0.\dot{6} \text{)} \] (e) \[ \frac{2^{x+2}-2^{x+3}}{2^{x+1}-2^{x+2}} = 2 \] (f) \[ \frac{9^{x}+3^{x+1}}{18^{x} \cdot 2^{1-x}} = \frac{1+3^{1-x}}{2} \] (g) \[ \frac{3 \cdot 2^{x}-2^{x-1}}{2^{x}+2^{x+2}} = \frac{1}{2} \quad \text{(or } 0.5 \text{)} \] (h) \[ \frac{4^{x}+2^{2 x-1}}{2^{2 x-1}} = 3 \] (i) \[ \frac{2 \cdot 3^{x+2}+3^{x-3}}{5 \cdot 3^{x-2}} = \frac{487}{15} \] If you need further assistance or explanations for any specific part, feel free to ask!