Estimate the nearest perfect squares around the number
Find which two perfect squares the number lies between
Use the square roots of those perfect squares as bounds
If the number is close to a perfect square, use a closer estimate
For large numbers, break the number into factors and simplify if possible
Use prime factorization to reduce square roots of perfect-square factors
Memorize common square roots such as 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
Use the average of nearby square roots as a rough estimate
Check your estimate by squaring it mentally
Refine the estimate by adjusting up or down until the square is close to the target number