Estimate the square root of 72

Question

Solve in your head. Estimate \sqrt{72} to three decimal places. Aim: under 20 seconds.

Accepted Answer

Use the ending-in-5 trick: 8.5^2 = 72.25 (from 85^2 = 7225). That is essentially on top of 72, far tighter than 64 or 81. Linearize downward from 72.25: \sqrt{72} \approx 8.5 - \frac{0.25}{2 	imes 8.5} = 8.5 - \frac{0.25}{17} \approx 8.5 - 0.0147 = 8.485. True value: 8.4853. Three decimals, no calculator. Takeaway. When the target sits just below a half-integer square (here 72 vs 72.25), the correction is tiny and the linearization is razor-sharp. Building the reflex of finding the nearest \dots.5 anchor doubles your usable set of square-root anchors for free.

Estimate the square root of 72

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