Quickly estimate 2^64 without using a pen/papar.

estimate 2^64

2^10=1024 ~10^3
2^64=(2^10)^6 * 2^4
=> (10^3)^6*16
=> 10^18*16
=> 1.6 * 10 ^ 19

= 16,000,000,000,000,000,000
Calculator says: 18,446,744,073,709,551,616


or:



2 ^ 10 = 1.024 * (10^3)
2 ^ 60 = (1.024 ^ 6) * (10 ^ 18)
2 ^ 64 = (16 * (1.024 ^ 6) * (10 ^ 18) )
All, we need to solve is 1.024 ^ 6. using binomial expansion, ignoring the smaller terms we get : (1 + 0.024) ^ 6 = 1 + 6 * 0.024 = 1.144 = 1.15 (approx)
Hence the answer is : (16 * 1.15) * (10 ^ 18) = 18.4 * (10 ^ 18)

It is much closer to the actual answer and very fast to calculate.