Simplify the following: \( \begin{array}{lll}\text { (1) } \frac{3^{2 x+1}}{3^{2 x}} & \text { (2) } \frac{2^{x+5}}{2^{x+2}} & \text { (3) } \frac{5^{m+4}}{5^{m+5}} \\ \text { (4) } \frac{2^{3 x+6}}{2^{3 x+4}} & \text { (5) } \frac{6^{4 x-2}}{6^{4 x-4}} & \text { (6) } \frac{4^{n}}{2^{2 n-3}} \\ \text { (7) } \frac{7^{2 x-1}}{49^{x-1}} & \text { (8) } \frac{9^{x+1}}{3^{2 x} \cdot 81} & \text { (9) } \frac{16^{p-3}}{8^{p-1}} \\ \text { (10) } \frac{5^{x} \cdot 25^{x-1}}{5 \cdot 125^{x}} & \text { (11) } \frac{9^{a} \cdot 4^{a-1}}{3^{2 a-1} \cdot 2^{2 a}} & \text { (12) } \frac{36^{x+2}}{6^{2 x+5}}\end{array} \)

Simplify the expression by following steps: - step0: Solution: \(\frac{4^{n}}{2^{2n-3}}\) - step1: Calculate: \(\frac{\left(2^{2}\right)^{n}}{2^{2n-3}}\) - step2: Calculate: \(\frac{2^{2n}}{2^{2n-3}}\) - step3: Calculate: \(2^{3}\) Calculate or simplify the expression \( \frac{36^{x+2}}{6^{2 x+5}} \). Simplify the expression by following steps: - step0: Solution: \(\frac{36^{x+2}}{6^{2x+5}}\) - step1: Calculate: \(\frac{\left(6^{2}\right)^{x+2}}{6^{2x+5}}\) - step2: Calculate: \(\frac{6^{2x+4}}{6^{2x+5}}\) - step3: Calculate: \(6^{-1}\) Calculate or simplify the expression \( \frac{16^{p-3}}{8^{p-1}} \). Simplify the expression by following steps: - step0: Solution: \(\frac{16^{p-3}}{8^{p-1}}\) - step1: Calculate: \(\frac{\left(2^{4}\right)^{p-3}}{8^{p-1}}\) - step2: Calculate: \(\frac{\left(2^{4}\right)^{p-3}}{\left(2^{3}\right)^{p-1}}\) - step3: Calculate: \(\frac{2^{4p-12}}{\left(2^{3}\right)^{p-1}}\) - step4: Calculate: \(\frac{2^{4p-12}}{2^{3p-3}}\) - step5: Calculate: \(2^{p-9}\) Calculate or simplify the expression \( \frac{7^{2 x-1}}{49^{x-1}} \). Simplify the expression by following steps: - step0: Solution: \(\frac{7^{2x-1}}{49^{x-1}}\) - step1: Calculate: \(\frac{7^{2x-1}}{\left(7^{2}\right)^{x-1}}\) - step2: Calculate: \(\frac{7^{2x-1}}{7^{2x-2}}\) - step3: Calculate: \(7\) Calculate or simplify the expression \( \frac{9^{x+1}}{3^{2 x} \cdot 81} \). Simplify the expression by following steps: - step0: Solution: \(\frac{9^{x+1}}{3^{2x}\times 81}\) - step1: Multiply: \(\frac{9^{x+1}}{3^{2x+4}}\) - step2: Factor the expression: \(\frac{3^{2x+2}}{3^{2x+4}}\) - step3: Divide the numbers: \(\frac{1}{3^{2x+4-\left(2x+2\right)}}\) - step4: Subtract the terms: \(\frac{1}{3^{2}}\) - step5: Evaluate the power: \(\frac{1}{9}\) Calculate or simplify the expression \( \frac{5^{x} \cdot 25^{x-1}}{5 \cdot 125^{x}} \). Simplify the expression by following steps: - step0: Solution: \(\frac{5^{x}\times 25^{x-1}}{5\times 125^{x}}\) - step1: Multiply by \(a^{-n}:\) \(\frac{5^{x}\times 25^{x-1}\times 5^{-1}}{125^{x}}\) - step2: Multiply: \(\frac{5^{3x-3}}{125^{x}}\) - step3: Factor the expression: \(\frac{5^{3x-3}}{5^{3x}}\) - step4: Divide the numbers: \(\frac{1}{5^{3x-\left(3x-3\right)}}\) - step5: Subtract the terms: \(\frac{1}{5^{3}}\) - step6: Evaluate the power: \(\frac{1}{125}\) Calculate or simplify the expression \( \frac{5^{m+4}}{5^{m+5}} \). Simplify the expression by following steps: - step0: Solution: \(\frac{5^{m+4}}{5^{m+5}}\) - step1: Multiply by \(a^{-n}:\) \(5^{m+4}\times 5^{-\left(m+5\right)}\) - step2: Calculate: \(5^{m+4}\times 5^{-m-5}\) - step3: Multiply: \(5^{m+4-m-5}\) - step4: Calculate: \(5^{-1}\) - step5: Rewrite the expression: \(\frac{1}{5}\) Calculate or simplify the expression \( \frac{3^{2 x+1}}{3^{2 x}} \). Simplify the expression by following steps: - step0: Solution: \(\frac{3^{2x+1}}{3^{2x}}\) - step1: Multiply by \(a^{-n}:\) \(3^{2x+1}\times 3^{-2x}\) - step2: Multiply: \(3^{2x+1-2x}\) - step3: Calculate: \(3\) Calculate or simplify the expression \( \frac{9^{a} \cdot 4^{a-1}}{3^{2 a-1} \cdot 2^{2 a}} \). Simplify the expression by following steps: - step0: Solution: \(\frac{9^{a}\times 4^{a-1}}{3^{2a-1}\times 2^{2a}}\) - step1: Factor the expression: \(\frac{3^{2a}\times 4^{a-1}}{3^{2a-1}\times 2^{2a}}\) - step2: Reduce the fraction: \(\frac{3\times 4^{a-1}}{2^{2a}}\) - step3: Factor the expression: \(\frac{3\times 2^{2a-2}}{2^{2a}}\) - step4: Reduce the fraction: \(\frac{3}{2^{2}}\) - step5: Evaluate the power: \(\frac{3}{4}\) Calculate or simplify the expression \( \frac{2^{x+5}}{2^{x+2}} \). Simplify the expression by following steps: - step0: Solution: \(\frac{2^{x+5}}{2^{x+2}}\) - step1: Multiply by \(a^{-n}:\) \(2^{x+5}\times 2^{-\left(x+2\right)}\) - step2: Calculate: \(2^{x+5}\times 2^{-x-2}\) - step3: Multiply: \(2^{x+5-x-2}\) - step4: Calculate: \(2^{3}\) - step5: Evaluate the power: \(8\) Calculate or simplify the expression \( \frac{2^{3 x+6}}{2^{3 x+4}} \). Simplify the expression by following steps: - step0: Solution: \(\frac{2^{3x+6}}{2^{3x+4}}\) - step1: Multiply by \(a^{-n}:\) \(2^{3x+6}\times 2^{-\left(3x+4\right)}\) - step2: Calculate: \(2^{3x+6}\times 2^{-3x-4}\) - step3: Multiply: \(2^{3x+6-3x-4}\) - step4: Calculate: \(2^{2}\) - step5: Evaluate the power: \(4\) Calculate or simplify the expression \( \frac{6^{4 x-2}}{6^{4 x-4}} \). Simplify the expression by following steps: - step0: Solution: \(\frac{6^{4x-2}}{6^{4x-4}}\) - step1: Multiply by \(a^{-n}:\) \(6^{4x-2}\times 6^{-\left(4x-4\right)}\) - step2: Calculate: \(6^{4x-2}\times 6^{-4x+4}\) - step3: Multiply: \(6^{4x-2-4x+4}\) - step4: Calculate: \(6^{2}\) - step5: Evaluate the power: \(36\) Here are the simplified results for each expression: 1. \( \frac{3^{2x+1}}{3^{2x}} = 3^{(2x+1) - 2x} = 3^{1} = 3 \) 2. \( \frac{2^{x+5}}{2^{x+2}} = 2^{(x+5) - (x+2)} = 2^{3} = 8 \) 3. \( \frac{5^{m+4}}{5^{m+5}} = 5^{(m+4) - (m+5)} = 5^{-1} = \frac{1}{5} \) 4. \( \frac{2^{3x+6}}{2^{3x+4}} = 2^{(3x+6) - (3x+4)} = 2^{2} = 4 \) 5. \( \frac{6^{4x-2}}{6^{4x-4}} = 6^{(4x-2) - (4x-4)} = 6^{2} = 36 \) 6. \( \frac{4^{n}}{2^{2n-3}} = \frac{(2^{2})^{n}}{2^{2n-3}} = \frac{2^{2n}}{2^{2n-3}} = 2^{(2n) - (2n-3)} = 2^{3} = 8 \) 7. \( \frac{7^{2x-1}}{49^{x-1}} = \frac{7^{2x-1}}{(7^{2})^{x-1}} = \frac{7^{2x-1}}{7^{2x-2}} = 7^{(2x-1) - (2x-2)} = 7^{1} = 7 \) 8. \( \frac{9^{x+1}}{3^{2x} \cdot 81} = \frac{(3^{2})^{x+1}}{3^{2x} \cdot 3^{4}} = \frac{3^{2x+2}}{3^{2x+4}} = 3^{(2x+2) - (2x+4)} = 3^{-2} = \frac{1}{9} \) 9. \( \frac{16^{p-3}}{8^{p-1}} = \frac{(2^{4})^{p-3}}{(2^{3})^{p-1}} = \frac{2^{4(p-3)}}{2^{3(p-1)}} = 2^{4p-12 - (3p-3)} = 2^{p-9} \) 10. \( \frac{5^{x} \cdot 25^{x-1}}{5 \cdot 125^{x}} = \frac{5^{x} \cdot (5^{2})^{x-1}}{5 \cdot (5^{3})^{x}} = \frac{5^{x} \cdot 5^{2x-2}}{5^{1} \cdot 5^{3x}} = \frac{5^{(x + 2x - 2)}}{5^{(1 + 3x)}} = 5^{(3x - 2) - (3x + 1)} = 5^{-3} = \frac{1}{125} \) 11. \( \frac{9^{a} \cdot 4^{a-1}}{3^{2a-1} \cdot 2^{2a}} = \frac{(3^{2})^{a} \cdot (2^{2})^{a-1}}{3^{2a-1} \cdot 2^{2a}} = \frac{3^{2a} \cdot 2^{2a-2}}{3^{2a-1} \cdot 2^{2a}} = \frac{3^{2a} \cdot 2^{2a-2}}{3^{2a-1} \cdot 2^{2a}} = 3^{(2a) - (2a-1)} \cdot 2^{(2a-2) - (2a)} = 3^{1} \cdot 2^{-2} = \frac{3}{4} \) 12. \( \frac{36^{x+2}}{6^{2x+5}} = \frac{(6^{2})^{x+2}}{6^{2x+5}} = \frac{6^{2x+4}}{6^{2x+5}} = 6^{(2x+4) - (2x+5)} = 6^{-1} \) In summary, the simplified results are: 1. \( 3 \) 2. \( 8 \) 3. \( \frac{1}{5} \) 4. \( 4 \) 5. \( 36 \) 6. \( 8 \) 7. \( 7 \) 8. \( \frac{1}{9} \) 9. \( 2^{p-9} \) 10. \( \frac{1}{125} \) 11. \( \frac{3}{4} \) 12. \( 6^{-1} \)