1. **Problem Statement:** Find the square roots of 100 and 169 using the method of repeated subtraction. 2. **Method Explanation:** The method of repeated subtraction to find the square root is based on the fact that the sum of the first $n$ odd numbers equals $n^2$. For example, $1 + 3 + 5 + \cdots + (2n-1) = n^2$. To find $\sqrt{x}$, we repeatedly subtract consecutive odd numbers from $x$ until the remainder is zero. The number of subtractions performed is the square root. 3. **Finding $\sqrt{100}$:** - Start with 100. - Subtract consecutive odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19. - Perform the subtractions: $$100 - 1 = 99$$ $$99 - 3 = 96$$ $$96 - 5 = 91$$ $$91 - 7 = 84$$ $$84 - 9 = 75$$ $$75 - 11 = 64$$ $$64 - 13 = 51$$ $$51 - 15 = 36$$ $$36 - 17 = 19$$ $$19 - 19 = 0$$ - Number of subtractions = 10, so $\sqrt{100} = 10$. 4. **Finding $\sqrt{169}$:** - Start with 169. - Subtract consecutive odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25. - Perform the subtractions: $$169 - 1 = 168$$ $$168 - 3 = 165$$ $$165 - 5 = 160$$ $$160 - 7 = 153$$ $$153 - 9 = 144$$ $$144 - 11 = 133$$ $$133 - 13 = 120$$ $$120 - 15 = 105$$ $$105 - 17 = 88$$ $$88 - 19 = 69$$ $$69 - 21 = 48$$ $$48 - 23 = 25$$ $$25 - 25 = 0$$ - Number of subtractions = 13, so $\sqrt{169} = 13$. **Final answers:** $$\sqrt{100} = 10$$ $$\sqrt{169} = 13$$