To find the numerical equivalent of \( 4^{9 \cdot 4^{-7}} \), we can simplify the expression step by step.
First, let's rewrite \( 4^{-7} \): \[ 4^{-7} = \frac{1}{4^7} \]
Now, we can rewrite \( 9 \cdot 4^{-7} \) as: \[ 9 \cdot 4^{-7} = \frac{9}{4^7} \]
Now calculate \( 4^{9 \cdot 4^{-7}} \): \[ 4^{9 \cdot 4^{-7}} = 4^{\frac{9}{4^7}} \]
Next, recall the laws of exponents which state \( a^{b^c} = a^{b \cdot c} \), so we can simplify this to: \[ 4^{\frac{9}{4^7}} = 4^{9} \cdot 4^{-7} \]
Thus, \[ 4^{9} \cdot 4^{-7} = 4^{9-7} = 4^{2} \]
Now we can translate \( 4^2 \) into its numerical form: \[ 4^2 = 16 \]
Therefore, the numerical equivalent of \( 4^{9 \cdot 4^{-7}} \) is: \[ \boxed{16} \]
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