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I have a question in my textbook that asks you to evaluate each statement as true or false, but I'm having trouble understanding the notation they're using.

$(1,2,3) \in \mathbb{Z}^5$

What does it mean when the set of integers is raised to the power of $5$?

Also, what would the P($\mathbb{Z}^5$) mean?

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The set $\mathbb{Z}^5$ consists of all 5-tuples of integers. Since $(1,2,3)$ is a 3-tuple, it doesn't belong to $\mathbb{Z}^5$, but rather to $\mathbb{Z}^3$.

For your other question, $P(S)$ is the power set of $S$, consisting of all subsets of $S$.

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    @dgx Note that it usually uses \mathcal font $\mathcal P(\mathbb Z)$ – 2017-02-25