Estimating square roots mentally
Bracket the number between perfect squares you know. √50 sits between √49 (7) and √64 (8), much closer to 7 — about 7.07. This estimate is accurate enough for quick checks.
Where square roots appear
Geometry (diagonals and the Pythagorean theorem), physics (distance fallen), finance (volatility) and statistics (standard deviation) all lean on square roots.