Power of Positive Real Number is Positive/Rational Number
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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.
$\blacksquare$