Proportion of Power
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Theorem
Let $x$ and $y$ be proportional.
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Let $n \in \Z$.
Then $x^n \propto y^n$.
Proof
Let $x \propto y$.
Then $\exists k \ne 0: x = k \times y$ by the definition of proportion.
Raising both sides of this equation to the $n$th power:
\(\ds x^n\) | \(=\) | \(\ds \paren {k \times y}^n\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds k^n \times y^n\) |
so $k^n$ is the desired constant of proportion.
The result follows from the definition of proportion.
$\blacksquare$