Sum of Indices of Real Number

Theorem

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

Positive Integers

Let $n, m \in \Z_{\ge 0}$ be positive integers.

Let $r^n$ be defined as $r$ to the power of $n$.

Then:

$r^{n + m} = r^n \times r^m$

Integers

Let $n, m \in \Z$ be integers.

Let $r^n$ be defined as $r$ to the power of $n$.

Then:

$r^{n + m} = r^n \times r^m$

Rational Numbers

Let $x, y \in \Q$ be rational numbers.

Let $r^x$ be defined as $r$ to the power of $n$.

Then:

$r^{x + y} = r^x \times r^y$