I've formally defined the power whose base is a positive number and its exponent is a real non-zero number like this:
$$x^{\alpha}=\sup\left\{x^p\in\mathbb{R}\mid p\in\mathbb{Q},0<p\leq\alpha\right\},\text{ if $\alpha>0$.}$$ $$x^{\alpha}=\inf\left\{x^p\in\mathbb{R}\mid p\in\mathbb{Q},\alpha\leq p<0\right\},\text{ if $\alpha<0$.}$$
I want to prove the properties which are true for rational numbers ($x>0,\alpha,\beta\neq 0$):
- $x^\alpha\cdot x^\beta=x^{\alpha+\beta}$.
- $(x^\alpha)^\beta=x^{\alpha\cdot\beta}$.
- $(x\cdot y)^\alpha=x^\alpha\cdot y^\alpha$.
Any ideas? My problem is that there're 2 supremum and I don't have any idea to handle that. I hope, I didn't commit any mistake (sorry for my English!).
If $A,B$ are nonempty sets of nonnegative real numbers then $(\sup A)(\sup B) = \sup (AB)$ and $(\inf A)(\inf B)=\inf (AB),$ where $AB = \{ ab \mid a\in A,\ b\in B \}$.
So for example, if $\alpha,\beta>0,$ then $$\begin{align} x^\alpha \cdot x^\beta &= \sup\{x^q \mid q\in\mathbb{Q},\ 0<q\leq\alpha\} \cdot \sup\{x^r \mid r\in\mathbb{Q},\ 0<r\leq\beta\} \\ &= \sup\{x^q\cdot x^r \mid q,r\in\mathbb{Q},\ 0<q\leq\alpha,\ 0<r\leq\beta\}) \\ &= \sup\{x^{q+r} \mid q,r\in\mathbb{Q},\ 0<q+r\leq\alpha+\beta\}) \\ &= \sup\{x^{s} \mid s\in\mathbb{Q},\ 0<s\leq\alpha+\beta\}) \\ &= x^{\alpha+\beta} . \end{align}$$