# Rational Power of Product of Real Numbers

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## Theorem

Let $r, s \in \R_{> 0}$ be (strictly) positive real numbers.

Let $x \in \Q$ be a rational number.

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

Then:

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

## Proof

Let $x = \dfrac p q$ where $p, q \in \Z$ and $q > 0$.

We have:

 $\ds r^x s^x$ $=$ $\ds \paren {r^p}^{1 / q} \paren {s^p}^{1 / q}$ $\ds$ $=$ $\ds \paren {r^p s^p}^{1 / q}$ $\ds$ $=$ $\ds \paren {\paren {r s}^p}^{1 / q}$ $\ds$ $=$ $\ds \paren {r s}^{p / q}$ $\ds$ $=$ $\ds \paren {r s}^x$

$\blacksquare$