Estimate the square root of 2000

Question

No calculator. Solve in your head. Estimate \sqrt{2000} to a couple of significant figures. Aim: under 15 seconds.

Accepted Answer

Factor out the largest easy perfect square: \sqrt{2000} = \sqrt{400 	imes 5} = 20\sqrt{5}. Use the anchor \sqrt{5} \approx 2.236: 20 	imes 2.236 = 44.7. So \sqrt{2000} \approx 44.7 (check: 44.7^2 = 1998). Takeaway. Split the number as (perfect square) 	imes (small leftover), take the clean root out front, and multiply by a memorised root of the leftover. Keep \sqrt{2} \approx 1.414, \sqrt{3} \approx 1.732, \sqrt{5} \approx 2.236 on call and most roots become one multiplication.

Estimate the square root of 2000

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