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.
For positive bases a, you have the general rule
This follows from the fact that exponentials and logarithms are inverses of each other, and that the logarithm has the property that
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,
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).