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How would you calculate any radical expression without a calculator? I know you can do something like long division for square roots, but I mean for big radical expressions, such as $\sqrt[36]{12345}$

PS: The reason I need this is that I want to code a calculator, and obviously, I cannot use a calculator in a calculator. I think Math SE is the best place to ask, let me know if I should ask on stackoverflow.

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  • $\begingroup$ Newton's approximation...? $$a_{n+1}=a_n-\frac{(a_n)^{36}-12345}{36(a_n)^{36-1}}$$ for this particular problem. In general, change all of the $36's$ to the root, and the $12345$ part to the number you are rooting. $\endgroup$ – Simply Beautiful Art Sep 16 '16 at 0:45
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In days of yore, we would do this with base-10 logs. In your example we'd write $\log \sqrt[36]{12345} = (1/36)\log(12345) =(1/36)4.09149 = .11365$. Then we'd take the antilog (which means using the log table backwards) and get $10^{0.11365} = 1.29913.$ I think the old calculators would grind out Taylor series for $\log$ and $10^x$, and more modern ones had tables in memory and would just look up the values.

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