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so I was asking myself whether you could calculate a number raised to the power of a rational number, e.g. 3^2.5, without a calculator.

My motiviation comes from looking at my calculator and what all those functions like sin, square root... look like. I like it when I know the background off all these. Let's me understand them better if you know what I mean.

I'm thanking for your answers in advance!!!

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    $\begingroup$ This is a good question! Of course, before calculators, people did compute these things, but mostly by tables—but then how were those tables assembled? I believe, internally, most calculators use tables combined with Taylor series to approximate near known points. Unfortunately, this is not suitable for MO, but I could imagine it doing well at MSE. $\endgroup$ – LSpice 2 days ago