# 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 $$\frac pq\in\mathbb Q$$.

Proofs: 1 Corollaries: 1

Proofs: 1 2

Github: ### References

#### Bibliography

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