定义　给定实数$a>0$，$a \ne 1$。设$x$为无理数，我们规定
$$a^x=\left\{ \begin{array}{l} \sup\limits_{r<x}\left\{\left. a^r \right|r为有理数 \right\},当a>1时\\ \inf\limits_{r<x}\left\{\left. a^r \right|r为无理数 \right\},当0<a<1时 \end{array} \right.$$

定理　设$a>0$，$\alpha$，$\beta$为任意实数，则有
$$a^\alpha \cdot a^\beta = a^{\alpha + \beta}，\left( a^\alpha \right)^\beta = a^{\alpha \beta}。$$