Geometric Sequence in Lowest Terms has Coprime Extremes/Proof 2

Theorem
A geometric progression of integers in lowest terms has extremes which are coprime.

Proof
Let $P$ be a geometric progression of natural numbers of length $n$.

Let the common ratio of $P$ be expressed in canonical form as $\dfrac p q$.

From Construction of Geometric Progression in Lowest Terms:
 * $P = \left({q^n, p q^{n - 1}, p^2 q^{n - 2}, \ldots, p^{n - 1} q, p^n}\right)$

By definition of canonical form:
 * $p \perp q$

It follows from Powers of Coprime Numbers are Coprime that:
 * $p^n \perp q^n$

Hence the result.