Estimate the square root of 80

Question

No calculator. Solve in your head. Estimate \sqrt{80} to two decimal places. Aim: under 20 seconds.

Accepted Answer

80 sits just below 81 = 9^2. Use the linear correction \sqrt{a^2 - d} \approx a - \dfrac{d}{2a} with a = 9, d = 1: \sqrt{80} \approx 9 - \frac{1}{18} \approx 9 - 0.056 = 8.944. Check: 8.944^2 = 79.99. (Cleanly, \sqrt{80} = 4\sqrt5 = 8.944.) Takeaway. Near a perfect square a^2, each unit you move away changes the root by about 	frac{1}{2a}. That single derivative, "one over twice the root", turns any nearby square root into a quick add or subtract.

Estimate the square root of 80

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