To evaluate \(4^{9 \cdot 4^{-7}}\), we first simplify the expression inside the exponent.
The exponent can be rewritten as: \[ 9 \cdot 4^{-7} = 9 \cdot \frac{1}{4^7} = \frac{9}{4^7} \]
Now, substituting this back into the original expression we get: \[ 4^{\frac{9}{4^7}} = 4^{9 \cdot 4^{-7}} = 4^{\frac{9}{4^7}} = 4^{9} \cdot 4^{-7} \]
Calculating that yields: \[ 4^{9 - 7} = 4^{2} \]
Thus, the numerical equivalent of the expression \(4^{9 \cdot 4^{-7}}\) simplifies to \(4^2\).
Therefore, the answer is: 4 squared.