Estimate the cube root of 100

Question

No calculator. Solve in your head. Estimate \sqrt[3]{100} to two decimal places. Aim: under 30 seconds.

Accepted Answer

Bracket it: 4^3 = 64 and 5^3 = 125, so the answer is between 4 and 5, nearer 5. Anchor at a = 4.6, where 4.6^3 = 97.3. The cube-root derivative says a change d in the input shifts the root by \dfrac{d}{3a^2}. Here 3a^2 = 3 	imes 21.16 \approx 63.5 and d = 100 - 97.3 = 2.7: \sqrt[3]{100} \approx 4.6 + \frac{2.7}{63.5} \approx 4.6 + 0.043 = 4.64. Check: 4.64^3 \approx 99.9. So \sqrt[3]{100} \approx 4.64. Takeaway. For cube roots, the correction factor is 	frac{1}{3a^2} (versus 	frac{1}{2a} for square roots). Start from a nearby value whose cube you know, then take one linear step toward the …

Estimate the cube root of 100

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