Refining Your Estimate

Bracketing gets you close; a little arithmetic gets you closer. Take √200: you know 14 squared is 196 and 15 squared is 225. Since 200 is 4 above 196, and the gap between 196 and 225 is 29, estimate 14 + 4/29 — about 14.14. The true value is 14.142, so you land within a hundredth.

This works for any root. The trick is simply assuming the values rise in a straight line between the two perfect squares. With practice, two-decimal accuracy from pure mental math takes only a few seconds, and it is a fast way to sanity-check a calculator before trusting it.