Those Exponents

Someone asked me how non-integer exponentials are calculated.
Like  the joke says, very carefully. They are a difficulty and anyone with a new idea
on the subject will be listened to. Current best practice is to convert th exponent to
a fraction. Like 5^(5/11). Now it is obvious one wants to 5^5 first and then,
that result ^(1/11) or in the opposite order; the end result will be the same. And to
the question as to how one is supposed to find exponent (1/11), or the eleventh root of a number,
best practice there is to use natural logarithms.

ln(x)^(5/11) = 5/11 ln(x)
One finds ln(5), then one multiplies that number by 5, and divides by 11. Voilà.
Logarithms were invented to simplify calculations and this is how they do it.

ln(5) is worth 1.609437912...; * (3/11), gives .731562687...
e^.731562687... gives 2.078325845...
A calculator will verify this answer.

How were ln tables arrived at to begin with. By arduous calculation (leading eventually to the
invention of the slide rule). Newton devised an approximation method.
Interestingly, e itself can be found by (1 + 1/x)^x for the largest x one can imagine.
It is a limit. e is also given by

1 + 1/2 + 1/3 + 1/4 ...etc and not those abominable prime numbers. Whole numbers form the
ordinal series, and all of a sudden, everyone is worried about their mathematical properties.
Get a life!!

http://www.mathsisfun.com/algebra/exponent-fractional.html