How big is 2 to the 40th?

Question

Solve in your head. Roughly how big is 2^{40}, and what is it exactly? Aim: estimate in under 10 seconds.

Accepted Answer

The estimate. Use 2^{10} \approx 10^3, four blocks of ten: 2^{40} = (2^{10})^4 \approx (10^3)^4 = 10^{12}. So 2^{40} is about a trillion. It is a bit high, because 1024 exceeds 1000 by about 2.4\%, and four blocks compound that: 1.024^4 \approx 1.10, giving roughly 1.1 	imes 10^{12}. The exact value. Build up from the anchor. 2^{20} = 1{,}048{,}576 and 2^{30} = 2^{20} 	imes 1024 = 1{,}073{,}741{,}824, so 2^{40} = 2^{30} 	imes 1024 = 1{,}073{,}741{,}824 	imes 1024 = 1{,}099{,}511{,}627{,}776. That is about 1.0995 	imes 10^{12}, right where the estimate said, just under 1.1 trillion. Takea…

How big is 2 to the 40th?

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