Given: \(A = \{x, y, z\}\)

Step 1: Find the number of subsets of \(A\):

If a set has \(n\) elements, its power set has \(2^n\) subsets.

\(|A| = 3 \Rightarrow |P(A)| = 2^3 = 8.\)

Step 2: Now we need the number of subsets of \(P(A)\), i.e. the power set of the power set.

So, \(|P(P(A))| = 2^{|P(A)|} = 2^8 = \boxed{256}.\)

✅ Final Answer: 256