Estimate 1.5 to the 10th power

Question

Solve in your head. Estimate (1.5)^{10} within about 2%. Aim: under 45 seconds.

Accepted Answer

Square-and-multiply, three squarings and one product: - 1.5^2 = 2.25 - 1.5^4 = 2.25^2 = 5.0625 (ending-in-5: 225^2 = 50625) - 1.5^8 = 5.0625^2 \approx 25.63 (linearize: 5.0625^2 \approx 25 + 2 	imes 5 	imes 0.0625 = 25.625) - 1.5^{10} = 1.5^8 	imes 1.5^2 \approx 25.63 	imes 2.25 Last product: 25.63 	imes 2.25 = 25.63 	imes 2 + 25.63 	imes 	frac14 = 51.26 + 6.41 \approx 57.7. True value: 57.665. The estimate lands within 0.1\%. Takeaway. Exponentiation by squaring turns n multiplications into \log2 n: the only skills consumed are squaring numbers near round anchors and one clean final …

Estimate 1.5 to the 10th power

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