Estimate the square root of 30

Question

Solve in your head. Estimate \sqrt{30} to three decimal places. Aim: under 20 seconds.

Accepted Answer

Half-integer squares are easy via the ending-in-5 trick: 5.5^2 = 30.25 (from 55^2 = 3025). That is far closer to 30 than either 25 or 36. Linearize downward from 30.25: \sqrt{30} \approx 5.5 - \frac{0.25}{2 	imes 5.5} = 5.5 - \frac{0.25}{11} \approx 5.5 - 0.0227 = 5.477. True value: 5.4772. Three decimals, no calculator. Takeaway. Every number ending in 5 has an instantly computable square, so a half-integer anchor (5.5^2 = 30.25) sits between consecutive perfect squares. Closer anchor means smaller correction means a more accurate one-step linearization.

Estimate the square root of 30

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