To simplify the expression \((14^3 \cdot 27^{-7})^0 / 3^4\), we first observe that anything raised to the power of 0 is 1 (as long as the base is not zero). Therefore, we have:
\[ (14^3 \cdot 27^{-7})^0 = 1 \]
Now substituting this into the expression, we get:
\[ \frac{1}{3^4} \]
Next, we calculate \(3^4\):
\[ 3^4 = 81 \]
So, the expression simplifies to:
\[ \frac{1}{81} \]
Thus, the final simplified result is:
\[ \frac{1}{81} \]