How to calculate a decimal power of a number

I wish to calculate a power like 2.142.14

When I ask my calculator to do it, I just get an answer, but I want to see the calculation.
So my question is, how to calculate this with a pen, paper and a bunch of brains.

Answer

For positive bases a, you have the general rule
ab=exp(bln(a))=eblna.

This follows from the fact that exponentials and logarithms are inverses of each other, and that the logarithm has the property that
ln(xr)=rln(x).

So you have, for example,
(2.14)2.14=eln((2.14)2.14)(because elnx=x)=e(2.14)ln(2.14)(because ln(xr)=rlnx)
Or more generally,
ab=eln(ab)=eblna.

In fact, this is formula can be taken as the definition of ab for a>0 and arbitrary exponent b (that is, not an integer, not a rational).

As to computing e2.14ln(2.14), there are reasonably good methods for approximating numbers like ln(2.14), and numbers like er (e.g., Taylor polynomials or other methods).

Attribution
Source : Link , Question Author : Mixxiphoid , Answer Author : Ross Millikan

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