One possible way to approximate the square root of 10 to three decimal places without using a calculator is:
- Find two perfect squares that are close to 10. For example, 9 and 16 are perfect squares, and 9 < 10 < 16.
- Find the square roots of these perfect squares. For example, the square root of 9 is 3, and the square root of 16 is 4.
- Use these values to make a guess for the square root of 10. For example, since 10 is closer to 9 than to 16, we can guess that the square root of 10 is closer to 3 than to 4. A reasonable guess could be 3.1 or 3.2.
- Check if the guess is too high or too low by squaring it and comparing it to 10. For example, if we guess 3.1, then 3.1^2 = 9.61, which is too low. If we guess 3.2, then 3.2^2 = 10.24, which is too high.
- Adjust the guess by changing the last decimal place up or down until the squared value is as close to 10 as possible without going over. For example, if we try 3.15, then 3.15^2 = 9.9225, which is still too low. If we try 3.16, then 3.16^2 = 9.9856, which is very close to 10 and within the desired accuracy of three decimal places.
Therefore, we can conclude that the square root of 10 is approximately 3.16.