1. **State the problem:** Calculate the value of $25 \times \sqrt{1127}$.\n\n2. **Formula and rules:** The square root function $\sqrt{x}$ gives the number which, when multiplied by itself, equals $x$. Multiplying by 25 means scaling the square root value by 25.\n\n3. **Calculate the square root:** $\sqrt{1127}$ is an irrational number, so we approximate it. Using a calculator, $\sqrt{1127} \approx 33.57$.\n\n4. **Multiply by 25:**\n$$25 \times \sqrt{1127} \approx 25 \times 33.57 = 839.25.$$\n\n5. **Final answer:**\n$$\boxed{839.25}.$$