Real Number to Negative Power/Integer
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Theorem
Let $r \in \R_{> 0}$ be a positive real number.
Let $n \in \Z$ be an integer.
Let $r^n$ be defined as $r$ to the power of $n$.
Then:
- $r^{-n} = \dfrac 1 {r^n}$
Proof
Let $n \in \Z_{\ge 0}$.
Then from Real Number to Negative Power: Positive Integer:
- $r^{-n} = \dfrac 1 {r^n}$
It remains to show that this holds when $n < 0$.
Let $n \in \Z_{<0}$.
Then $n = - m$ for some $m \in \Z_{> 0}$.
Thus:
\(\ds r^{-m}\) | \(=\) | \(\ds \dfrac 1 {r^m}\) | Real Number to Negative Power: Positive Integer | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \dfrac 1 {r^{-m} }\) | \(=\) | \(\ds \dfrac 1 {1 / r^m}\) | taking reciprocal of both sides | ||||||||||
\(\ds \) | \(=\) | \(\ds r^m\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \dfrac 1 {r^n}\) | \(=\) | \(\ds r^{-n}\) |
$\blacksquare$