To simplify the expression \( 2^4 \cdot 11^{-5} \cdot 11^7 \), we can start by simplifying the powers of \( 11 \).
Recall that the property of exponents states that when multiplying with the same base, you add the exponents:
\[ 11^{-5} \cdot 11^7 = 11^{-5 + 7} = 11^2 \]
Now, we can rewrite the entire expression:
\[ 2^4 \cdot 11^{-5} \cdot 11^7 = 2^4 \cdot 11^2 \]
Next, we calculate the values of \( 2^4 \) and \( 11^2 \):
\[ 2^4 = 16 \] \[ 11^2 = 121 \]
Now, substituting these values back into the expression:
\[ 2^4 \cdot 11^2 = 16 \cdot 121 \]
Next, we multiply \( 16 \) by \( 121 \):
\[ 16 \cdot 121 = 1936 \]
Thus, the final answer is:
\[ \boxed{1936} \]