Real Power of Strictly Positive Real Number is Strictly Positive

Theorem

Let $x$ be a strictly positive real number.

Let $y$ be a real number.

Then:

$x^y > 0$

where $x^y$ denotes $x$ raised to the $y$th power.

Proof

From the definition of power:

$x^y = \exp \left({y \ln x}\right)$

From Exponential of Real Number is Strictly Positive:

$x^y = \exp \left({y \ln x}\right) > 0$

$\blacksquare$