Estimate the cube root of 500 to three significant figures in your head, and give a second method that confirms the third figure.

Estimate the cube root of 500 to three significant figures in your head, and give a second method that confirms the third figure.

Approach: Anchor on the nearest perfect cube, factor it out, and expand the remaining ratio to second order, then confirm with one iteration of the cube root recurrence.

7.94. Anchor on 8^3 = 512, so the cube root of 500 is 8 times the cube root of 500/512, which is 8 times the cube root of 1 - 0.0234375. The binomial expansion of (1 + u) to the power one third is 1 + u/3 - u^2/9, giving 1 - 0.0078125 - 0.000061 = 0.9921265, so the answer is 8 * 0.9921265 = 7.93701. The true value is 7.937005, so to three significant figures the answer is 7.94. A Newton iteration confirms it independently, since x maps to (2x + 500/x^2)/3, and starting at x = 8 gives (16 + 7.8125)/3 = 7.9375. That is already correct to four figures, and both methods agree on 7.937, so the rounded 7.94 is safe.

Follow-up: How does the anchor change when you estimate the cube root of 5000, and what answer does it give to three significant figures?

Key concepts: cube root, binomial expansion, Newton iteration.