I 10^(1/10) is VERY close to 2^(1/3)

AI Thread Summary
The discussion centers on the numerical proximity of 10^(1/10) and 2^(1/3), which are approximately 1.2599 and 1.2589, respectively. While the values are close when rounded to three decimal places, the consensus is that this similarity is largely coincidental rather than indicative of a deeper mathematical relationship. Attempts to factor the numbers did not yield additional insights into their closeness. The conversation highlights that while they may seem interchangeable in certain contexts, they are not mathematically equivalent. Overall, the closeness of these numbers is acknowledged but deemed not significant beyond coincidence.
Juanda
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Is there a subjacent reason that explains why these two numbers are so close?
$$10^{1/10} \approx 2^{1/3}$$

For context, this is where I found out about this.
Source: https://www.instarengineering.com/p...ration_Testing_of_Small_Satellites_Part_5.pdf
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Is it just a coincidence? I tried factoring the numbers but it doesn't provide any additional information. From the infinite number of possibilities, how did they realize that ##10^{1/10} \approx 2^{1/3}## are so close together that it is acceptable to use the ##2^{1/3}## without losing too much accuracy?
 
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##2^{10}=1024, \sqrt[3] {1024} \approx 10##
 
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How did I not see it? 🤦‍♂️
Thank you. It's clear now.
 
Juanda said:
Is it just a coincidence?
Pretty much -- ##10^{1/10}## is not all that close to ##2^{1/3}##.
The first is ~1.2599, and the second is ~1.2589. Rounding to 3 decimal places gives 1.260 vs. 1.259. I guess these are close if you consider ##\sqrt 2 \approx 1.4## as they did in the quoted article.
 
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