(c) Simplify the following: \( \begin{array}{lll}\text { (1) } \frac{3^{2 x+1}}{3^{2 x}} & \text { (2) } \frac{2^{x+5}}{2^{x+2}} & \text { (3) } \frac{5^{n+4}}{5^{m+5}} \\ \text { (4) } \frac{2^{3 x+5}}{2^{3 x+9}} & \text { (5) } \frac{6^{4 x-2}}{6^{4 x-4}} & \text { (6) } \frac{4^{n}}{2^{2 n-3}} \\ \text { (7) } \frac{7^{2 x-1}}{49^{x-1}} & \text { (8) } \frac{9^{x+1}}{3^{2 x} \cdot 81} & \text { (9) }-\frac{16^{p-3}}{8^{p-4}} \\ \text { (10) } \frac{5^{x} \cdot 25^{x-1}}{5 \cdot 125^{x}} & \text { (11) } \frac{9^{a} \cdot 4^{a-1}}{3^{2 a-1} \cdot 2^{2 a}} & \text { (12) } \frac{36^{x+2}}{6^{2 x+5}} \\ \text { (13) } \frac{50^{x+1}}{2^{x+1}} 25^{x+2} & \text { (14) } \frac{5 \cdot 45^{y}}{9^{y} \cdot 5^{y+2}} & \text { (15) } \frac{18^{x} \cdot 8^{x-1}}{9^{x+1} \cdot 4^{2 x-1}}\end{array} \)

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 \( \frac{9^{a} \cdot 4^{a-1}}{3^{2 a-1} \cdot 2^{2 a}} \). 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 \( \frac{18^{x} \cdot 8^{x-1}}{9^{x+1} \cdot 4^{2 x-1}} \). Simplify the expression by following steps: - step0: Solution: \(\frac{18^{x}\times 8^{x-1}}{9^{x+1}\times 4^{2x-1}}\) - step1: Factor the expression: \(\frac{9^{x}\times 2^{x}\times 8^{x-1}}{9^{x+1}\times 4^{2x-1}}\) - step2: Reduce the fraction: \(\frac{2^{x}\times 8^{x-1}}{9\times 4^{2x-1}}\) - step3: Factor the expression: \(\frac{2^{x}\times 8^{x-1}}{9\times 2^{4x-2}}\) - step4: Reduce the fraction: \(\frac{8^{x-1}}{9\times 2^{3x-2}}\) - step5: Factor the expression: \(\frac{2^{3x-3}}{9\times 2^{3x-2}}\) - step6: Reduce the fraction: \(\frac{1}{9\times 2}\) - step7: Calculate: \(\frac{1}{18}\) Calculate or simplify the expression \( \frac{5^{n+4}}{5^{m+5}} \). Simplify the expression by following steps: - step0: Solution: \(\frac{5^{n+4}}{5^{m+5}}\) - step1: Multiply by \(a^{-n}:\) \(5^{n+4}\times 5^{-\left(m+5\right)}\) - step2: Calculate: \(5^{n+4}\times 5^{-m-5}\) - step3: Multiply: \(5^{n+4-m-5}\) - step4: Calculate: \(5^{n-1-m}\) 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 \( \frac{3^{2 x+1}}{3^{2 x}} \). 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 \( \frac{6^{4 x-2}}{6^{4 x-4}} \). 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: \(36\) Calculate or simplify the expression \( \frac{4^{n}}{2^{2 n-3}} \). 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 \( \frac{2^{3 x+5}}{2^{3 x+9}} \). Simplify the expression by following steps: - step0: Solution: \(\frac{2^{3x+5}}{2^{3x+9}}\) - step1: Multiply by \(a^{-n}:\) \(2^{3x+5}\times 2^{-\left(3x+9\right)}\) - step2: Calculate: \(2^{3x+5}\times 2^{-3x-9}\) - step3: Multiply: \(2^{3x+5-3x-9}\) - step4: Calculate: \(2^{-4}\) - step5: Express with a positive exponent: \(\frac{1}{2^{4}}\) - step6: Evaluate the power: \(\frac{1}{16}\) Calculate or simplify the expression \( \frac{50^{x+1}}{2^{x+1} \cdot 25^{x+2}} \). Simplify the expression by following steps: - step0: Solution: \(\frac{50^{x+1}}{2^{x+1}\times 25^{x+2}}\) - step1: Factor the expression: \(\frac{2^{x+1}\times 25^{x+1}}{2^{x+1}\times 25^{x+2}}\) - step2: Reduce the fraction: \(\frac{25^{x+1}}{25^{x+2}}\) - step3: Divide the numbers: \(\frac{1}{25^{x+2-\left(x+1\right)}}\) - step4: Subtract the terms: \(\frac{1}{25^{1}}\) - step5: Simplify: \(\frac{1}{25}\) 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 \( -\frac{16^{p-3}}{8^{p-4}} \). Simplify the expression by following steps: - step0: Solution: \(-\frac{16^{p-3}}{8^{p-4}}\) - step1: Calculate: \(-2^{p}\) 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 \( \frac{5 \cdot 45^{y}}{9^{y} \cdot 5^{y+2}} \). Simplify the expression by following steps: - step0: Solution: \(\frac{5\times 45^{y}}{9^{y}\times 5^{y+2}}\) - step1: Multiply by \(a^{-n}:\) \(\frac{5\times 45^{y}\times 5^{-\left(y+2\right)}}{9^{y}}\) - step2: Calculate: \(\frac{5\times 45^{y}\times 5^{-y-2}}{9^{y}}\) - step3: Multiply: \(\frac{5^{-1-y}\times 45^{y}}{9^{y}}\) - step4: Factor the expression: \(\frac{5^{-1-y}\times 9^{y}\times 5^{y}}{9^{y}}\) - step5: Reduce the fraction: \(5^{-1-y}\times 5^{y}\) - step6: Express with a positive exponent: \(\frac{1}{5^{1+y}}\times 5^{y}\) - step7: Rewrite the expression: \(\frac{5^{y}}{5^{1+y}}\) - step8: Calculate: \(5^{y-\left(1+y\right)}\) - step9: Calculate: \(5^{-1}\) - step10: Express with a positive exponent: \(\frac{1}{5}\) 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: \(\frac{1}{125}\) 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. \( \frac{5^{n+4}}{5^{m+5}} = 5^{(n+4) - (m+5)} = 5^{n-1-m} \) 4. \( \frac{2^{3x+5}}{2^{3x+9}} = 2^{(3x+5) - (3x+9)} = 2^{-4} = \frac{1}{16} \) 5. \( \frac{6^{4x-2}}{6^{4x-4}} = 6^{(4x-2) - (4x-4)} = 6^{2} = 36 \) 6. \( \frac{4^{n}}{2^{2n-3}} = \frac{(2^2)^{n}}{2^{2n-3}} = \frac{2^{2n}}{2^{2n-3}} = 2^{3} = 8 \) 7. \( \frac{7^{2x-1}}{49^{x-1}} = \frac{7^{2x-1}}{(7^2)^{x-1}} = \frac{7^{2x-1}}{7^{2(x-1)}} = 7^{(2x-1) - 2(x-1)} = 7^{2x-1-2x+2} = 7^{1} = 7 \) 8. \( \frac{9^{x+1}}{3^{2x} \cdot 81} = \frac{(3^2)^{x+1}}{3^{2x} \cdot 3^4} = \frac{3^{2(x+1)}}{3^{2x} \cdot 3^4} = \frac{3^{2x+2}}{3^{2x+4}} = 3^{(2x+2) - (2x+4)} = 3^{-2} = \frac{1}{9} \) 9. \( -\frac{16^{p-3}}{8^{p-4}} = -\frac{(2^4)^{p-3}}{(2^3)^{p-4}} = -\frac{2^{4(p-3)}}{2^{3(p-4)}} = -2^{4p-12 - 3p + 12} = -2^{p} \) 10. \( \frac{5^{x} \cdot 25^{x-1}}{5 \cdot 125^{x}} = \frac{5^{x} \cdot (5^2)^{x-1}}{5 \cdot (5^3)^{x}} = \frac{5^{x} \cdot 5^{2x-2}}{5^{1+3x}} = \frac{5^{x + 2x - 2}}{5^{3x + 1}} = \frac{5^{3x - 2}}{5^{3x + 1}} = 5^{(3x - 2) - (3x + 1)} = 5^{-3} = \frac{1}{125} \) 11. \( \frac{9^{a} \cdot 4^{a-1}}{3^{2a-1} \cdot 2^{2a}} = \frac{(3^2)^{a} \cdot (2^2)^{a-1}}{3^{2a-1} \cdot 2^{2a}} = \frac{3^{2a} \cdot 2^{2a-2}}{3^{2a-1} \cdot 2^{2a}} = \frac{3^{2a} \cdot 2^{2a-2}}{3^{2a-1} \cdot 2^{2a}} = \frac{3^{2a}}{3^{2a-1}} \cdot \frac{2^{2a-2}}{2^{2a}} = 3^{1} \cdot 2^{-2} = \frac{3}{4} \) 12. \( \frac{36^{x+2}}{6^{2x+5}} = \frac{(6^2)^{x+2}}{6^{2x+5}} = \frac{6^{2(x+2)}}{6^{2x+5}} = 6^{2x+4 - (2x+5)} = 6^{-1} = \frac{1}{6} \) 13. \( \frac{50^{x+1}}{2^{x+1} \cdot 25^{x+2}} = \frac{(2 \cdot 25)^{x+1}}{2^{x+1} \cdot (5^2)^{x+2}} = \frac{2^{x+1} \cdot 5^{2(x+1)}}{2^{x+1} \cdot 5^{2x+4}} = \frac{5^{2(x+1)}}{5^{2x+4}} = 5^{2x+2 - (2x+4)} = 5^{-2} = \frac{1}{25} \) 14. \( \frac{5 \cdot 45^{y}}{9^{y} \cdot 5^{y+2}} = \frac{5 \cdot (5 \cdot 9)^{y}}{9^{y} \cdot 5^{y+2}} = \frac{5^{1+y} \cdot 9^{y}}{9^{y} \cdot 5^{y+2}} = \frac{5^{1+y}}{5^{y+2}} = 5^{1+y - (y+2)} = 5^{-1} = \frac{1}{5} \) 15. \( \frac{18^{x} \cdot 8^{x-1}}{9^{x+1} \cdot 4^{2x-1}} = \frac{(2 \cdot 9)^{x} \cdot (2^3)^{x-1}}{(3^2)^{x+1} \cdot (2^2)^{2x-1}} = \frac