Estimate the cube root of 30

Question

Solve in your head. Estimate \sqrt[3]{30} to three decimal places. Aim: under 15 seconds.

Accepted Answer

The linearization idea extends to any root. For cube roots, linearize around the nearest perfect cube: \sqrt[3]{a^3 + \varepsilon} \approx a + \frac{\varepsilon}{3a^2}. The nearest cube to 30 is 27 = 3^3, so a = 3 and \varepsilon = 3: \sqrt[3]{30} \approx 3 + \frac{3}{3 \cdot 9} = 3 + \frac{3}{27} = 3 + 0.111 = 3.111. True value: \sqrt[3]{30} \approx 3.10723. The estimate is high by about 0.004, again a slight overshoot because x^{1/3} is concave and the tangent line lies above the curve. The general rule comes from the derivative of x^{1/n}: \frac{d}{dx}x^{1/n} = \frac{1}{n}x^{1/n - 1}, whic…

Estimate the cube root of 30

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