Powers of Elements of Geometric Sequence are in Geometric Sequence

Theorem
Let $P = \left\langle{a_j}\right\rangle_{1 \mathop \le j \mathop \le n}$ be a geometric progression of integers.

Then the sequence $Q = \left\langle{b_j}\right\rangle_{1 \mathop \le j \mathop \le n}$ defined as:
 * $\forall j \in \left\{{1, 2, \ldots, n}\right\}: b_j = a_j^m$

where $m \in \Z_{>0}$, is a geometric progression.

Proof
From Form of Geometric Progression of Integers, the $j$th term of $P$ is given by:
 * $a_j = k q^{j - 1} p^{n - j}$

Thus the $j$th term of $Q$ is given by:
 * $b_j = k^m \left({q^m}\right)^{j - 1} \left({p^m}\right)^{n - j}$

From Form of Geometric Progression of Integers, this is a geometric progression.