# 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.

$\blacksquare$