1. **Problem Statement:** Find the square roots of 1000 and 169 using the method of repeated subtraction. 2. **Method Explanation:** The method of repeated subtraction for square roots involves subtracting consecutive odd numbers starting from 1 until the number reduces to zero or a negative number. The count of subtractions performed before reaching zero or negative is the square root. 3. **Formula/Rule:** The sum of the first $n$ odd numbers is $n^2$. So, if we can subtract $1, 3, 5, 7, \ldots$ repeatedly from a number until it reaches zero, the number of subtractions is the square root. 4. **Finding $\sqrt{1000}$:** - Start subtracting odd numbers from 1000: $1000 - 1 = 999$ $999 - 3 = 996$ $996 - 5 = 991$ $\ldots$ - Continue this process until the remainder is less than zero. - Count the number of subtractions. - After 31 subtractions, the remainder becomes negative, so $\sqrt{1000} \approx 31$. 5. **Finding $\sqrt{169}$:** - Start subtracting odd numbers from 169: $169 - 1 = 168$ $168 - 3 = 165$ $165 - 5 = 160$ $\ldots$ - Continue until remainder is zero. - After 13 subtractions, remainder is zero, so $\sqrt{169} = 13$. 6. **Summary:** - $\sqrt{1000} \approx 31$ - $\sqrt{169} = 13$