To solve the equation
\[ 3^{-2} \cdot \frac{3^{p}}{ } = 3^{4}, \]
we can first simplify the left side using the properties of exponents.
The product of powers property states that \( a^m \cdot a^n = a^{m+n} \). Therefore,
\[ 3^{-2} \cdot 3^{p} = 3^{-2 + p}. \]
Now, we have:
\[ 3^{-2 + p} = 3^{4}. \]
Since the bases are the same, we can equate the exponents:
\[ -2 + p = 4. \]
To solve for \( p \), add 2 to both sides:
\[ p = 4 + 2 = 6. \]
Thus, the solution is
\[ \boxed{6}. \]