Estimate 1.1 to the 10th power

Question

Solve in your head. Estimate (1.1)^{10} within about 2%. Aim: under 40 seconds.

Accepted Answer

Square-and-multiply: three squarings plus one product. - 1.1^2 = 1.21 - 1.1^4 = 1.21^2 = 1.4641 (ending-in-1 squares are easy: 121^2 = 14641) - 1.1^8 = 1.4641^2 \approx 2.14 (linearize: 1.46^2 = 2.1316, nudged up to about 2.144) - 1.1^{10} = 1.1^8 	imes 1.1^2 \approx 2.144 	imes 1.21 Last product: 2.144 	imes 1.21 = 2.144 + 2.144 	imes 0.21 = 2.144 + 0.450 = 2.594. True value: 2.5937. The estimate is essentially exact. A useful cross-check: the rule of 72 says money at 10% doubles in about 72/10 = 7.2 years, so ten years should be a bit past a double, and 2.59 fits. Takeaway. Ten years of…

Estimate 1.1 to the 10th power

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