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

To simplify the expression \( \frac{5^{2 n}-5^{2 n+2}}{25^{n+1} \cdot 5^{2 n}} \), we can start by rewriting \( 25 \) as \( 5^2 \). Thus, \[ 25^{n+1} = (5^2)^{n+1} = 5^{2(n+1)} = 5^{2n + 2}. \] Now, substituting \( 25^{n+1} \) into the expression gives: \[ \frac{5^{2 n}-5^{2 n + 2}}{5^{2n + 2} \cdot 5^{2 n}}. \] Next, let's simplify the numerator: \[ 5^{2 n} - 5^{2 n + 2} = 5^{2 n} - 5^{2 n} \cdot 5^2 = 5^{2 n}(1 - 25). \] So the numerator becomes: \[ 5^{2 n}(-24) = -24 \cdot 5^{2 n}. \] Now, substituting this back into the expression, we have: \[ \frac{-24 \cdot 5^{2 n}}{5^{2 n + 2} \cdot 5^{2 n}} = \frac{-24 \cdot 5^{2 n}}{5^{4 n + 2}} = \frac{-24}{5^{4 n + 2 - 2 n}} = \frac{-24}{5^{2 n + 2}}. \] Now, it’s in a simplified form: \[ \frac{-24}{5^{2 n + 2}}. \] Thus, the final simplified expression is: \[ \boxed{\frac{-24}{5^{2(n+1)}}}. \]