Power of Positive Real Number is Positive

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 $r$.