To solve the expression \( 12^{-5} \cdot 12^{7} \), we will apply the product rule of integer exponents, which states that when multiplying two expressions with the same base, you can add their exponents. The rule can be written as:
\[ a^m \cdot a^n = a^{m+n} \]
In this case, our base \( a \) is \( 12 \), \( m \) is \( -5 \), and \( n \) is \( 7 \).
Using the product rule, we can combine the exponents:
\[ 12^{-5} \cdot 12^{7} = 12^{-5 + 7} \]
Now, we perform the addition of the exponents:
\[ -5 + 7 = 2 \]
Thus, we have:
\[ 12^{-5} \cdot 12^{7} = 12^{2} \]
Next, we can calculate \( 12^{2} \):
\[ 12^{2} = 12 \cdot 12 = 144 \]
So, the numerical equivalent of \( 12^{-5} \cdot 12^{7} \) is:
\[ \boxed{144} \]
In summary, we applied the product rule of integer exponents to combine the exponents of the same base and calculated the resulting exponent to find the final numerical value.