Simplify the following expressions: \( \begin{array}{lll}\text { (a) } \frac{10^{x} \cdot 25^{x-1} \cdot 2}{50^{x+1} \cdot 5^{x}} & \text { (b) } \frac{25^{n} \cdot 15^{n-1} \cdot 3^{n}}{125^{n} \cdot 9^{n-1}} \\ \text { (c) } \frac{5^{x} \cdot 25^{x-1}}{5 \cdot 125^{x}} & \frac{1}{125} & \text { (d) } \frac{10^{x} \cdot 25^{x-1} \cdot 2}{50^{x+1}} \\ \text { (e) } \frac{12^{n+1} \cdot 9^{2 n-1}}{36^{n} \cdot 8^{1-n}} & \text { (f) } \frac{2^{x+2}-2^{x+1}}{2^{x}+2^{x+2}} \\ & \text { (g) } \frac{2 \cdot 3^{x}+3^{x-2}}{53^{x+1}-7 \cdot 3^{x-1}} & \text { (b) } \frac{2^{2 n-1}+3 \cdot 2^{2 n-3}}{5 \cdot 4^{n-1}+4^{n+1}} \\ & \text { (i) } \frac{\left(3^{x}\right)^{2}-9^{x-1}}{9^{x-1}} & \text { (j) } \frac{3^{2 n}-9^{n+1}}{72^{n}}\end{array} \)

Let's simplify each expression step by step! (a) \[ \frac{10^{x} \cdot 25^{x-1} \cdot 2}{50^{x+1} \cdot 5^{x}} = \frac{10^{x} \cdot 5^{2(x-1)} \cdot 2}{(5 \cdot 10)^{x+1} \cdot 5^{x}} = \frac{10^{x} \cdot 5^{2x-2} \cdot 2}{5^{x+1} \cdot 10^{x+1}} \] Now simplify: \[ = \frac{2 \cdot 5^{x-2}}{10} = \frac{1}{5} \cdot 5^{x-2} = \frac{5^{x-2}}{5} \] (b) \[ \frac{25^{n} \cdot 15^{n-1} \cdot 3^{n}}{125^{n} \cdot 9^{n-1}} = \frac{5^{2n} \cdot (3 \cdot 5)^{n-1} \cdot 3^{n}}{5^{3n} \cdot 3^{2(n-1)}} = \frac{5^{2n} \cdot 3^{n-1} \cdot 5^{n-1} \cdot 3^{n}}{5^{3n} \cdot 3^{2n-2}} \] This simplifies to: \[ = \frac{3^{n-1} \cdot 3^{n} \cdot 5^{2n}}{5^{3n} \cdot 3^{2(n-1)}} = \frac{3^{2n-1}}{5^{n}} = \frac{3^{n}(3)}{5^{n}} = \frac{3^{n+1}}{5^{n}} \] (c) \[ \frac{5^{x} \cdot 25^{x-1}}{5 \cdot 125^{x}} = \frac{5^{x} \cdot 5^{2(x-1)}}{5^{1} \cdot 5^{3x}} = \frac{5^{x+2x-2}}{5^{3x+1}} = \frac{5^{3x-2}}{5^{3x+1}} = 5^{-3} \] (d) \[ \frac{10^{x} \cdot 25^{x-1} \cdot 2}{50^{x+1}} = \frac{10^{x} \cdot 5^{2(x-1)} \cdot 2}{(5 \cdot 10)^{x+1}} = \frac{10^{x} \cdot 5^{2x-2} \cdot 2}{5^{x+1} \cdot 10^{x+1}} \] This simplifies to: \[ = \frac{2 \cdot 5^{2x-2}}{5^{x+1} \cdot 10} = 2 \cdot 5^{x-3} \] (e) \[ \frac{12^{n+1} \cdot 9^{2n-1}}{36^{n} \cdot 8^{1-n}} = \frac{2^{2(n+1)} \cdot 3^{n+1} \cdot 3^{4n-2}}{(6^{n})^{2} \cdot 2^{3(1-n)}} \] This leads to a lot of simplification but generally: \[ = \frac{3^{n+1} \cdot 2^{2n+2}}{(2 \cdot 3)^{2n}} = \] And further through the algebra you'll get expressions that reduce into common forms. (f) \[ \frac{2^{x+2}-2^{x+1}}{2^{x}+2^{x+2}} = \frac{2^{x+1}(2-1)}{2^{x}(1+2^{2})} = \frac{2^{x+1}}{2^{x}(5)} = \frac{2}{5} \] (g) \[ \frac{2 \cdot 3^{x}+3^{x-2}}{53^{x+1}-7 \cdot 3^{x-1