Irrational exponent understanding

Question

For an integer number $a$

$$x^a=\{(x)(x)(x)...(x)\} (a\,times)$$ $$x^{\frac{1}{b}}=n\rightarrow\;\{(n)(n)(n)...(n)\}(b\,times)=x$$

For rational number $m=\frac{a}{b}$

$$x^m=x^\frac{a}{b}=(x^a)^\frac{1}{b}$$

And can be though of as a combination of the situations before

What about

$$x^e$$

How would one calculate or picture this from more basic operations?

Answer 1

For any $a\in\Bbb R$ and any positive $x$, one has by definition $$x^a=e^{a\ln x}$$ While this could seem to be a loopy definition, it is actually not since $e^x$ is primarily defined not via exponentiation, but via one of the two equivalent definitions:

1. $$e^x=\sum_{i=0}^\infty \frac{x^i}{i!}$$

2. $f:x\mapsto e^x$ is the only function $\Bbb R\to\Bbb R$ such that $f(0)=1$ and for all $x\in\Bbb R$, $f'(x)=f(x).$

Answer 2

$x^e$ is the limit of the sequence

$$x^2, x^{27/10}, x^{271/100}, x^{2718/1000}, \cdots$$

By the way. This is a conceptual not a computational definition. No one would want to compute the thousandth root of $x^{2718}$ by hand. Especially since the sequence will converge to $x^e$ very slowly.

According to Wolfram alpha, to the first ten digits...

$$5^e \approx 79.43235917 $$

$$5^{2718/1000} \approx 79.39633798 $$

$$\text{absolute error $= |5^e - 5^{2718/1000}| \approx 0.036$}$$

$$\text{relative error $= 100 \dfrac{|5^e - 5^{2718/1000}|}{5^e} \approx 0.045\%$}$$