To solve the equation \( (8^2)p = 8^4 \), we can first rewrite the left side:
\[ (8^2)^p = 8^{2p} \]
Now, we equate both sides:
\[ 8^{2p} = 8^4 \]
Since the bases are the same, we can set the exponents equal to each other:
\[ 2p = 4 \]
Now, divide both sides by 2 to solve for \( p \):
\[ p = \frac{4}{2} = 2 \]
The value of \( p \) is \( \boxed{2} \).