Estimate the square root of 90

Question

Solve in your head. Estimate \sqrt{90} to three decimal places. Aim: under 20 seconds.

Accepted Answer

Use the ending-in-5 trick: 9.5^2 = 90.25 (from 95^2 = 9025). This anchor is a quarter above 90, dramatically closer than 81 or 100. Linearize downward from 90.25: \sqrt{90} \approx 9.5 - \frac{0.25}{2 	imes 9.5} = 9.5 - \frac{0.25}{19} \approx 9.5 - 0.0132 = 9.487. True value: 9.4868. Three decimals, no calculator. Takeaway. For any target just under a half-integer square, the answer is (half-integer) minus (gap)/(twice the half-integer). The larger the anchor, the smaller the correction term, so big roots like \sqrt{90} come out especially accurate.

Estimate the square root of 90

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