To simplify the expression \(2(4)^2 \cdot 2(2)^2\), we first rewrite it in a clearer format:
\[ 2^4 \cdot 2^2 \]
Next, we apply the Product Rule of Exponents, which states that when multiplying two expressions with the same base, you add the exponents:
\[ 2^4 \cdot 2^2 = 2^{4 + 2} = 2^6 \]
Now we can simplify \(2^6\):
\[ 2^6 = 64 \]
Therefore, the simplified expression is:
\[ \boxed{64} \]