1. **State the problem:** Simplify the expression $25\sqrt{1127}$. 2. **Recall the formula and rules:** The square root of a product can be written as the product of square roots: $$\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$$ 3. **Factorize 1127 to simplify the square root:** Check if 1127 has any perfect square factors. - 1127 divided by 13: $1127 \div 13 = 86.69$ (not exact) - 1127 divided by 17: $1127 \div 17 = 66.29$ (not exact) - 1127 divided by 7: $1127 \div 7 = 161$ (not exact) Try 1127 divided by 49 (since $7^2=49$): $1127 \div 49 = 23$ exactly. So, $1127 = 49 \times 23$. 4. **Rewrite the expression:** $$25 \sqrt{1127} = 25 \sqrt{49 \times 23}$$ 5. **Use the property of square roots:** $$25 \sqrt{49 \times 23} = 25 \times \sqrt{49} \times \sqrt{23}$$ 6. **Simplify the square root of 49:** $$\sqrt{49} = 7$$ 7. **Multiply constants:** $$25 \times 7 = 175$$ 8. **Final simplified expression:** $$175 \sqrt{23}$$ **Answer:** $175 \sqrt{23}$