Definition:Power (Algebra)/Real Number/Definition 3

Definition
Let $x \in \R_{>0}$ be a (strictly) positive real number.

Let $r \in \R$ be a real number.

First let $x > 1$.

Let $r$ be expressed by its decimal expansion:
 * $r = n \cdotp d_1 d_2 d_3 \ldots$

For $k \in \Z_{> 0}$, let $\psi_1, \psi_2 \in \Q$ be rational numbers defined as:

Then $x^r$ is defined as the (strictly) positive real number $\xi$ defined as:
 * $\displaystyle \lim_{k \mathop \to \infty} x^{\psi_1} \le \xi \le x^{\psi_2}$

In this context, $x^{\psi_1}, x^{\psi_2}$ denote $x$ to the rational powers $\psi_1$ and $\psi_2$.

Next let $x < 1$.

Then $x^r$ is defined as:
 * $x^r := \left({\dfrac 1 x}\right)^{-r}$

Finally, when $x = 1$:


 * $x^r = 1$

Also see

 * Equivalence of Definitions of Number to Real Power

The difference between the two extremes is:
 * $x^{n + \frac {d_1} {10} + \cdots + \frac {d_k} {10^k} } \left({x^{\frac 1 {10^k} } - 1}\right)$

and by taking $k$ large enough, it is possible to reach any accuracy for $x^r$ as is neeeded.