(related to Proposition: Rational Powers of Positive Numbers)
Let \(a > 0\) be a positive real number. The exponential function of general base \(a\) with the rational exponent \(\frac pq\in\mathbb Q\), (\(q\neq 0\)), is well-defined, since it is defined for all real numbers \(x\in\mathbb R\), and every rational number \(\frac pq\) is also a real number.
It remains to be shown that indeed \[a^{\frac pq}=\sqrt[q]{a^p}=\exp_a\left(\frac pq\right).\quad\quad( * )\] For an integer \(p\in\mathbb Z\), the \(p\)-th power function has been identified with the exponential function of general base to the exponent \(p\): \[a^p=\exp_a(p).\quad\quad( * * )\] Note that for \(q\in\{-1,1\}\), \( ( * ) \) and \( ( * * ) \) are identical. For all integers \(q\), with \(|q|\ge 2\), the \(q\)-th root of the positive base \(b\) is defined as an inverse function to the \(q\)-th power function: \[\sqrt[q] b\quad \text{is inverse to}\quad b^q\quad\text{for all integers }q\text{ with }|q|\ge 2.\] It follows from \( ( * * ) \) and for \(b=a^p\) that \[a^p=\exp_a(p)=\exp_a\left(q\cdot \frac pq\right)=\left(\exp_a\left(\frac pq\right)\right)^q,\] which is equivalent to \( ( * ) \).
