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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 welldefined and justifies the definition of the rational power function of positive real numbers \(a\to a^{\frac pq}\):
\[a^{\frac pq}=\sqrt[q]{a^p}:=\exp_a\left(\frac pq \right).\]
Notes
 $a^{\frac pq}$ can also be written as the generalized power of $a$, i.e. as $$a^{\frac pq}=\exp_a\left(\frac pq\right).$$
 If \(\frac pq\) is a rational number \(\ge 0\), then the function is defined for all \(a \in\mathbb R\).
 If \(\frac pq\) is a rational number \(< 0\), then the function is defined for positive bases \(a > 0\) only.
The following interactive figure demonstrates the rational power of different real bases \(a \in\mathbb R\) for different rational exponents \(\frac pq\in\mathbb Q\).
Table of Contents
Proofs: 1 Corollaries: 1
Mentioned in:
Proofs: 1 2
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References
Bibliography
 Forster Otto: "Analysis 1, Differential und Integralrechnung einer VerĂ¤nderlichen", Vieweg Studium, 1983