1. **State the problem:** Solve the equation \(5^{2n} \times 25^{2n-1} = 625\). 2. **Recall the rules and formulas:** - Express all terms with the same base if possible. - Use the property \(a^m \times a^n = a^{m+n}\). - Recognize that \(25 = 5^2\) and \(625 = 5^4\). 3. **Rewrite the equation with base 5:** \[ 5^{2n} \times (5^2)^{2n-1} = 5^4 \] 4. **Simplify the powers:** \[ 5^{2n} \times 5^{2(2n-1)} = 5^4 \] \[ 5^{2n} \times 5^{4n - 2} = 5^4 \] 5. **Combine the exponents:** \[ 5^{2n + 4n - 2} = 5^4 \] \[ 5^{6n - 2} = 5^4 \] 6. **Set the exponents equal since the bases are the same:** \[ 6n - 2 = 4 \] 7. **Solve for \(n\):** \[ 6n = 4 + 2 \] \[ 6n = 6 \] \[ n = \frac{6}{6} = 1 \] **Final answer:** \(n = 1\)