Final answer:
The square roots of 100 and 169 can be found by repeatedly subtracting consecutive odd numbers starting with 1. After subtracting 10 odd numbers from 100, we reach zero, indicating the square root is 10. Similarly, for 169, we subtract 13 odd numbers to reach zero, indicating the square root is 13.
Step-by-step explanation:
Finding the square roots of 100 and 169 through repeated subtraction involves repeatedly subtracting consecutive odd numbers starting from 1 until you reach zero. For the number 100, you subtract 1, then 3, then 5, and so on, keeping track of how many subtractions you perform:
- 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
Since there are 10 numbers that we have subtracted, including 19 itself, the square root of 100 is 10.
Similarly, to find the square root of 169, we perform the same method:
- 169 - 1 = 168
- 168 - 3 = 165
- 165 - 5 = 160
- .... Keep subtracting until...
- 4 - 25 = -21 (which means we overshot, so we don't count this step)
You continue this process, and once you have subtracted the number 23, you will reach 0. This happens on the 13th subtraction, which tells you that the square root of 169 is 13.