Simplify \( \frac{25^{2 n-4}}{5^{3 n+1} \cdot 5^{2 n-3} \cdot 5} \)

To simplify the expression \( \frac{25^{2 n-4}}{5^{3 n+1} \cdot 5^{2 n-3} \cdot 5} \), we first rewrite \( 25 \) as \( 5^2 \). Thus, we have: \[ 25^{2n-4} = (5^2)^{2n-4} = 5^{2(2n-4)} = 5^{4n-8}. \] Now substituting this back into the expression, we get: \[ \frac{5^{4n-8}}{5^{3n+1} \cdot 5^{2n-3} \cdot 5}. \] Next, we can simplify the denominator by combining the exponents: \[ 5^{3n+1} \cdot 5^{2n-3} \cdot 5 = 5^{(3n+1) + (2n-3) + 1} = 5^{3n + 2n + 1 - 3 + 1} = 5^{5n - 1}. \] Putting this all together, we have: \[ \frac{5^{4n-8}}{5^{5n-1}}. \] Using the property of exponents \(\frac{a^m}{a^n} = a^{m-n}\), we can simplify further: \[ 5^{(4n-8) - (5n-1)} = 5^{4n - 8 - 5n + 1} = 5^{-n - 7}. \] Thus, the simplified form of the expression is: \[ \frac{1}{5^{n+7}}. \]