Estimate 1.4 to the 10th power

Question

Solve in your head. Estimate (1.4)^{10} within about 3%. Aim: under 50 seconds.

Accepted Answer

Square-and-multiply: three squarings plus one product. - 1.4^2 = 1.96 (call it "just under 2") - 1.4^4 = 1.96^2 \approx 3.84 (linearize near 2: 1.96^2 = 4 - 2 	imes 2 	imes 0.04 + \dots \approx 3.8416) - 1.4^8 = 3.8416^2 \approx 14.76 (linearize: 3.84^2 = 14.7456) - 1.4^{10} = 1.4^8 	imes 1.4^2 \approx 14.76 	imes 1.96 Last product: 14.76 	imes 1.96 = 14.76 	imes 2 - 14.76 	imes 0.04 = 29.52 - 0.59 = 28.93. True value: 28.925. The estimate lands within 0.1\%. The "1.96 is nearly 2" framing does the heavy lifting: squaring numbers just under a round anchor is a subtract-a-small-correcti…

Estimate 1.4 to the 10th power

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