What is 2 to the 18th, exactly?

Question

Solve in your head. What is 2^{18}, exactly? Aim: under 15 seconds.

Accepted Answer

Anchor on 2^{10} = 1024: 2^{18} = 2^{10} 	imes 2^{8} = 1024 	imes 256 = 262{,}144. To do 1024 	imes 256: 1000 	imes 256 = 256{,}000 and 24 	imes 256 = 6{,}144, summing to 262{,}144. The squaring route: 2^{18} = (2^9)^2 = 512^2. Square around 500: (500 + 12)^2 = 250{,}000 + 12{,}000 + 144 = 262{,}144. Same answer. For estimation: 2^{18} \approx 2.6 	imes 10^5, since 2^{10} \approx 10^3 and 2^8 = 256. Takeaway. Any even power of two is a perfect square of a smaller power (2^{18} = 512^2), and any power at all splits as 2^{10} times a small power. Keep both tricks handy and pick whichever …

What is 2 to the 18th, exactly?

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