Finding the Square Root of 289 by Repeated Subtraction Method

The repeated subtraction method is an old but effective technique for finding square roots. It involves repeatedly subtracting consecutive odd numbers from the given number until it reaches zero. The number of times the odd numbers are subtracted is the square root of the given number. Here's how to find the square root of 289 using this method:

Step 1: Write down the number whose square root you want to find (in this case, 289)

Step 2: Start with the number 1 and subtract consecutive odd numbers from it until you reach a negative number. Keep a count of how many times you subtracted an odd number. For example:

1 - 1 = 0 (count = 1)
3 - 1 = 2 (count = 2)
5 - 1 = 4 (count = 3)
7 - 1 = 6 (count = 4)
9 - 1 = 8 (count = 5)
11 - 1 = 10 (count = 6)
13 - 1 = 12 (count = 7)
15 - 1 = 14 (count = 8)
17 - 1 = 16 (count = 9)
19 - 1 = 18 (count = 10)
21 - 1 = 20 (count = 11)
23 - 1 = 22 (count = 12)
25 - 1 = 24 (count = 13)
27 - 1 = 26 (count = 14)
29 - 1 = 28 (count = 15)
31 - 1 = 30 (count = 16)
33 - 1 = 32 (count = 17)
35 - 1 = 34 (count = 18)
37 - 1 = 36 (count = 19)

Step 3: The count is the square root of the number whose square root you wanted to find. In this case, the count is 19, so the square root of 289 is 19.

Conclusion: Using the repeated subtraction method, we found that the square root of 289 is 19. This method can be a bit tedious for larger numbers, but it's a simple and effective way to find square roots without a calculator.