Sum of Indices of Real Number/Rational Numbers

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

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

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


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


Let $x = \dfrac p q, y = \dfrac u v$.


\(\ds r^\paren {x + y}\) \(=\) \(\ds r^\paren {\paren {p / q} + \paren {u / v} }\)
\(\ds \) \(=\) \(\ds r^\paren {\paren {p v + u q} / q v}\)
\(\ds \) \(=\) \(\ds \paren {r^\paren {1 / q v} }^\paren {p v + u q}\) Definition of Rational Power
\(\ds \) \(=\) \(\ds \paren {r^\paren {1 / q v} }^\paren {p v} \times \paren {r^\paren {1 / q v} }^\paren {u q}\) Sum of Indices of Real Number: Integers
\(\ds \) \(=\) \(\ds r^\paren {p v / q v} \times r^\paren {u q / q v}\) Definition of Rational Power
\(\ds \) \(=\) \(\ds r^\paren {p / q} \times r^\paren {u / v}\)
\(\ds \) \(=\) \(\ds r^x \times r^y\)