# Power of Positive Real Number is Positive/Rational Number

## Theorem

Let $x \in \R_{>0}$ be a (strictly) positive real number.

Let $q \in \Q$ be a rational number.

Then:

$x^q > 0$

where $x^q$ denotes the $x$ to the power of $q$.

## Proof

Let $q = \dfrac r s$, where $r \in \Z$, $s \in \Z \setminus \set 0$.

Then:

 $\ds x > 0$ $\leadsto$ $\ds x^r > 0$ Power of Positive Real Number is Positive: Integer $\ds$ $\leadsto$ $\ds \sqrt [s] {\paren {x^r} } > 0$ Existence of Positive Root of Positive Real Number $\ds$ $\leadsto$ $\ds x^{r / s}$ Definition of Rational Power

Hence the result.

$\blacksquare$