How to Extract the Cube Root of Two in your Head

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2 = 128/64 ~ 125/64.
Therefore 2^(1/3) ~ 5/4 = 1.250.
The above error in 2 is -3/64, therefore
The proportional error in 2 is -(3/64)/(128/64) = -3/128. Therefore
The proportional error in the cube root is one third,
Namely -(3/128)(1/3) = -1/128. Therefore
The absolute error in the cube root is approximately -(1/128)(1.250) ~ -0.010.
Adding back the absolute error of the first approximation:
2^(1/3) = 1.250 + 0.010 = 1.260.

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