Power of Positive Real Number is Positive
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Theorem
Natural Number
Let $x \in \R_{>0}$ be a (strictly) positive real number.
Let $n \in \Z_{\ge 0}$ be a positive integer.
Then:
- $x^n > 0$
where $x^n$ denotes the $n$th power of $x$.
Integer
Let $x \in \R_{>0}$ be a (strictly) positive real number.
Let $n \in \Z$ be an integer.
Then:
- $x^n > 0$
where $x^n$ denotes the $n$th power of $x$.
Rational Number
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 $q$.
Real Number
Let $x \in \R_{>0}$ be a (strictly) positive real number.
Let $r \in \R$ be a real number.
Then:
- $x^r > 0$
where $x^r$ denotes the $x$ to the power of $r$.