Estimate the square root of 50

Question

Solve in your head. Estimate \sqrt{50} to three decimal places. Aim: under 10 seconds.

Accepted Answer

Linearize around the nearest perfect square, 49: \sqrt{a^2 + \varepsilon} \approx a + \frac{\varepsilon}{2a}. With a = 7, \varepsilon = 1: \sqrt{50} \approx 7 + \frac{1}{14} = 7.0714\ldots True value: \sqrt{50} = 5\sqrt{2} \approx 7.0711. The linear step is off by only 0.0004, the tangent line overshoots slightly because \sqrt{x} is concave. Takeaway. \sqrt{a^2 + \varepsilon} \approx a + \frac{\varepsilon}{2a} is the workhorse for mental roots, and it always errs slightly high. Cross-check here: \sqrt{50} = 5\sqrt 2 and \sqrt 2 \approx 1.41421, giving 7.07107, consistent. Knowing \sqrt2, \sqr…

Estimate the square root of 50

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