To apply the Product Rule of Exponents, which states that \( a^m \cdot a^n = a^{m+n} \), we can first express the given expression \((1/2)^{2} \cdot (1/2)^{2}\) in terms of exponents.
Here, both terms share the same base, which is \(1/2\), and the exponents are both \(2\). Using the Product Rule, we combine the exponents:
\[ (1/2)^{2} \cdot (1/2)^{2} = (1/2)^{2 + 2} \]
Now we can simplify the exponent:
\[ 2 + 2 = 4 \]
Thus, we can write:
\[ (1/2)^{2} \cdot (1/2)^{2} = (1/2)^{4} \]
In exponential simplest form, the exponent is:
\[ \boxed{4} \]