Estimate 1.2 to the 10th power

Question

Solve in your head. Estimate (1.2)^{10} within about 2%. Aim: under 45 seconds.

Accepted Answer

Square-and-multiply: three squarings plus one product. - 1.2^2 = 1.44 - 1.2^4 = 1.44^2 \approx 2.07 (compute 1.44^2 = 2.0736) - 1.2^8 = 2.0736^2 \approx 4.30 (linearize: 2.07^2 = 4.2849, and the small extra pushes it to about 4.30) - 1.2^{10} = 1.2^8 	imes 1.2^2 \approx 4.30 	imes 1.44 Last product: 4.30 	imes 1.44 = 4.30 + 4.30 	imes 0.44 = 4.30 + 1.89 = 6.19. True value: 6.1917. The estimate lands within 0.1\%. Takeaway. Exponentiation by squaring turns ten multiplications into three squarings and a product. Worth caching for growth problems: 1.2^{10} \approx 6.19, so "20% a year for a …

Estimate 1.2 to the 10th power

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