Estimate 0.9 to the 10th power

Question

Solve in your head. Estimate (0.9)^{10} within about 2%. Aim: under 40 seconds.

Accepted Answer

Square-and-multiply: three squarings plus one product. - 0.9^2 = 0.81 - 0.9^4 = 0.81^2 = 0.6561 (81^2 = 6561) - 0.9^8 = 0.6561^2 \approx 0.4305 (linearize: 0.656^2 = 0.4303) - 0.9^{10} = 0.9^8 	imes 0.9^2 \approx 0.4305 	imes 0.81 Last product: 0.4305 	imes 0.81 = 0.4305 - 0.4305 	imes 0.19 = 0.4305 - 0.0818 = 0.3487. True value: 0.3487. The estimate is spot on. Sanity check against the naive answer: "lose 10% ten times" is not a 100\% loss. Because each cut is on a smaller base, you keep about 35\%, a reminder that percentage losses compound gently, never straight-line to zero. Takeaway.…

Estimate 0.9 to the 10th power

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