# Proof

(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 $$( * )$$.

Github: ### References

#### Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983