Square Roots and Powers in Your Head for Trading Tests

Estimating square roots from the nearest perfect square and memorizing key powers of 2 turns slow calculations into fast, defensible approximations.

Estimating square roots and computing powers in your head both rely on the same core idea: start from the nearest number you already know exactly, then correct for the small distance between that number and the one you actually need. A trading test rarely asks for a square root to six decimal places, it asks for an estimate accurate enough to make a pricing decision inside a few seconds, and a one-step correction from a known square gets there almost every time.

Any two-digit number squares by splitting it into tens and units and expanding the square of a sum. Twenty three squared is 20 plus 3, squared, which expands to 20 squared plus 2 times 20 times 3 plus 3 squared: 400 plus 120 plus 9, giving 529. The three pieces are each easy on their own, and adding three numbers is faster than the standard long multiplication of 23 by itself.

The same expansion handles larger two-digit numbers just as directly. Forty seven squared is 40 plus 7, squared, which is 1,600 plus 560 plus 49, giving 2,209. Recognizing that the middle term is always 2 times the tens part times the units part is the only piece of the pattern that needs to be memorized, since the two outer terms are just squares of numbers you already know.

The square root of 50 sits just above 7, since 7 squared is 49, one below 50. The correction adds the remaining 1 divided by twice the base, 14, giving roughly 0.071, for an estimate of 7.071. The actual value is 7.0711, so the one-step correction lands within a few thousandths, accurate enough for any pricing or sizing decision a trading test would ask for.

The square root of 90 works the same way from 9, since 9 squared is 81, nine below 90. The correction adds 9 divided by twice 9, which is 18, giving 0.5, for an estimate of 9.5 against an actual value of 9.4868, a small overshoot that comes from the correction being a linear approximation of a curve. The method is more accurate the closer the starting square is to the target number, so picking the nearest perfect square rather than any nearby one is what matters.