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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 well-defined 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

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