To estimate a square root without a calculator, determine the two perfect squares that come before and after the given number. Then using the square roots of the perfect squares as guides, estimate the given number’s square root based on its placement between the two perfect squares.
For example, 34 is between the perfect squares of 25 (5 x 5) and 36 (6 x 6). Therefore, 34 is approximately 5.8, as 34 is closer to 36 than to 25. For a more accurate estimation, divide the given number (34) by the approximation (5.8). Average this answer (approximately 5.862) with the approximation (5.8) to yield a closer estimation of the square root of the given number (approximately 5.831). Continue this process until the desired accuracy of the estimate is reached.