(2) \( \frac{10^{x} \cdot 25^{x+1}}{5^{x} \cdot 50^{x-1}} \)

To simplify the expression \( \frac{10^{x} \cdot 25^{x+1}}{5^{x} \cdot 50^{x-1}} \), let's break down each base into its prime factorization: 1. \( 10^x = (2 \cdot 5)^x = 2^x \cdot 5^x \) 2. \( 25^{x+1} = (5^2)^{x+1} = 5^{2(x+1)} = 5^{2x + 2} \) 3. \( 5^x \) remains as it is. 4. \( 50^{x-1} = (5 \cdot 10)^{x-1} = (5 \cdot 2 \cdot 5)^{x-1} = (5^2 \cdot 2)^{x-1} = 5^{2(x-1)} \cdot 2^{x-1} = 5^{2x - 2} \cdot 2^{x - 1} \) Now substitute these back into the original expression: \[ \frac{2^x \cdot 5^{x} \cdot 5^{2x + 2}}{5^{x} \cdot (5^{2x - 2} \cdot 2^{x - 1})} \] This simplifies to: \[ \frac{2^x \cdot 5^{x + 2x + 2}}{5^x \cdot 5^{2x - 2} \cdot 2^{x - 1}} = \frac{2^x \cdot 5^{3x + 2}}{5^{3x - 2} \cdot 2^{x - 1}} \] Now combine the exponents for both the \(5\) and \(2\): \[ = \frac{2^x \cdot 5^{3x + 2}}{5^{3x - 2} \cdot 2^{x - 1}} = 5^{(3x + 2) - (3x - 2)} \cdot 2^{x - (x - 1)} \] This results in: \[ = 5^{4} \cdot 2^{1} \] Therefore, the simplified form of the expression is: \[ 10^4 = 10000 \]