To find the numerical equivalent of \( 12^{-5} \cdot 12^7 \), we can apply the product rule of integer exponents, which states that when multiplying two powers with the same base, we can add their exponents. That is:
\[ a^m \cdot a^n = a^{m+n} \]
In our case, the base \( a \) is \( 12 \), \( m = -5 \), and \( n = 7 \).
Now we can apply the product rule:
\[ 12^{-5} \cdot 12^7 = 12^{-5 + 7} \]
Next, we calculate the exponent:
\[ -5 + 7 = 2 \]
So, we can rewrite the expression as:
\[ 12^{-5} \cdot 12^7 = 12^{2} \]
Now we can calculate \( 12^2 \):
\[ 12^2 = 144 \]
Thus, the numerical equivalent of \( 12^{-5} \cdot 12^7 \) is \( 144 \).
In summary:
- We applied the product rule of exponents to combine the terms: \( 12^{-5} \cdot 12^7 = 12^{-5 + 7} = 12^{2} \).
- We evaluated \( 12^{2} \) to get \( 144 \).
Final answer: \( 144 \)
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