Estimate the fifth root of 250

Question

No calculator. Solve in your head. Estimate \sqrt[5]{250} to two decimal places. Aim: under 30 seconds.

Accepted Answer

Anchor on 3^5 = 243, which is very close to 250. For an nth root, a small fractional change in the input produces about 	frac{1}{n} of that fractional change in the root. The input is up by 	frac{250 - 243}{243} \approx 2.9\%, so the fifth root rises by about 	frac{2.9\%}{5} \approx 0.58\%: \sqrt[5]{250} \approx 3 	imes (1 + 0.0058) \approx 3.017. Check: 3.017^5 \approx 250. Takeaway. The general rule: (1 + x)^{1/n} \approx 1 + 	frac{x}{n} for small x. Find the nearest known perfect power, express the gap as a percentage, and divide that percentage by the root's order. Works for any root…

Estimate the fifth root of 250

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