To find the square root of 152.7696, we can use a calculator or apply a method like the following:
1. Start by making pairs of digits from the right: 15, 27, 69, 6.
2. Find the largest integer whose square is less than or equal to the first pair, which is 15. Since 3^2 = 9 < 15 and 4^2 = 16 > 15, the integer we are looking for is 3.
3. Write the digit 3 as the first digit of the square root.
4. Subtract the square of 3 from 15 to get 6.
5. Bring down the next pair of digits, 27, and append them to 6 to get 627.
6. Double the first digit of the current root estimate (which is 3) to get 6.
7. Find the largest digit to fill in the blank in "3_ × _ = 6" such that the resulting product is less than or equal to 627. This digit is 7, since 3×7 = 21 < 627 and 3×8 = 24 > 627.
8. Write down the digit 7 as the second digit of the root estimate.
9. Subtract the square of 37 from 627 to get 60.
10. Bring down the next pair of digits, 69, and append them to 60 to get 6069.
11. Double the first digit of the current root estimate (which is 37) to get 74.
12. Find the largest digit to fill in the blank in "37_ × _ = 606" such that the resulting product is less than or equal to 6069. This digit is 1, since 37×1 = 37 < 6069 and 37×2 = 74 > 6069.
13. Write down the digit 1 as the third digit of the root estimate.
14. Subtract the square of 371 from 6069 to get 152.769.
Therefore, the square root of 152.7696 is approximately 12.36 (rounded to two decimal places).