In order to calculate the square root of a non-perfect square number, first find two perfect squares between which the number lies. Second, divide the number by one of the two square roots. Thirdly, take the average of the result and the root to discover the answer.
For example, to find the square root of 10, first note that it lies between 3 and 4, the square roots of 9 and 16, respectively. Then, divide 10 by 3 to get 3.33. Average 3.33 and 3 to get the answer 3.1667.
To come up with a more accurate answer, repeat the second and third steps. First, divide 10 by 3.1667 to get 3.1579. Then, average 3.1579 and 3.1667 to get 3.1623. Lastly, test the answer 3.1623 by squaring it. It should equal 10.0001.
To calculate the square root of a number by hand, one can use a variety of approaches. If the goal is to simplify the square root, you could, for example, change the square root of 50 into 5 times the square root of 2. Reducing the number to its simplest terms is always a good solution to finding the square root of a number.
Going further, by hand, involves estimation first. In the previous example, 5 times the square root of 2 can be estimated by multiplying 5 by 1.41, which is the square root of 2, to get approximately 7.1. For a larger number, you could find the square root using prime factorization.
For example, to find the square root of 2,000, you could find the square roots of 500 and 4, which are multiplied together to equal 2,000. The square root of 4 is 2, and the square root of 500 is 5 times the square root of 20, which is 10 times the square root of 5. Overall, the square root of 2,000 is then 20 times the square root of 5, or approximately 40.