TITLE: How to calculate a decimal power of a number
QUESTION [51 upvotes]: I wish to calculate a power like $$2.14 ^ {2.14}$$
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.

REPLY [5 votes]: we can find $2.14 ^{2.14}$ using basic arithmetic operations +,-,/,*.
Use binomial theorem
for rational number $n and $-1

$(1+x)^n =1+nx+n(n-1)x^2/2!+....$
note that in left hand side the power n is a fractional number but in the right hand side the powers are integers. that is, in the right hand side,  each term can be calculated using basic operations +,-,*,/. 
outline of the problem 
$2.14^{2.14}=(1.14+1)^{2.14}$
$=(1.14^{2.14}) * (1+1/1.14)^{2.14}$
$=(1+0.14)^{2.14}  *  (1+1/1.14)^{2.14}$
using binomial theorem two times (5 decimal places) and multiplying we get the answer