Product of Indices of Real Number/Rational Numbers

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Theorem

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 $x$.


Then:

$\paren {r^x}^y = r^{x y}$


Proof

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

Consider $\paren {\paren {r^x}^y}^{q v}$.

Then:

\(\ds \paren {\paren {r^x}^y}^{q v}\) \(=\) \(\ds \paren {\paren {r^\paren {p / q} }^\paren {u / v} }^{q v}\)
\(\ds \) \(=\) \(\ds \paren {\paren {\paren {r^\paren {p / q} }^\paren {u / v} }^v}^q\) Product of Indices of Real Number: Integers
\(\ds \) \(=\) \(\ds \paren {\paren {r^\paren {p / q} }^u}^q\) Definition of Rational Power
\(\ds \) \(=\) \(\ds \paren {r^\paren {p / q} }^{q u}\) Product of Indices of Real Number: Integers
\(\ds \) \(=\) \(\ds \paren {\paren {r^\paren {p / q} }^q}^u\) Product of Indices of Real Number: Integers
\(\ds \) \(=\) \(\ds \paren {r^p}^u\) Definition of Rational Power
\(\ds \) \(=\) \(\ds r^{p u}\) Product of Indices of Real Number: Integers
\(\ds \leadsto \ \ \) \(\ds \paren {\paren {r^\paren {p / q} }^\paren {u / v} }\) \(=\) \(\ds r^{\paren {p u} / \paren {q v} }\) taking $q v$th root of both sides
\(\ds \leadsto \ \ \) \(\ds \paren {r^x}^y\) \(=\) \(\ds r^{x y}\)

$\blacksquare$


Sources

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