To apply the product rule of integer exponents, we use the rule that states:
\[ a^m \cdot a^n = a^{m+n} \]
In this case, we have:
\[ 12^{-15} \cdot 12^{7} \]
Here, \(a = 12\), \(m = -15\), and \(n = 7\). According to the product rule:
\[ 12^{-15} \cdot 12^{7} = 12^{-15 + 7} = 12^{-8} \]
Next, we can find the numerical equivalent of \(12^{-8}\). Remember that a negative exponent indicates the reciprocal:
\[ 12^{-8} = \frac{1}{12^{8}} \]
Now, we can compute \(12^{8}\):
\[ 12^8 = (12^4)^2 \]
Calculating \(12^4\):
\[ 12^4 = 20736 \]
Now, squaring \(12^4\):
\[ 12^8 = 20736^2 \]
Calculating \(20736^2\):
\[ 20736^2 = 429981696 \]
Thus,
\[ 12^{-8} = \frac{1}{429981696} \]
So the numerical equivalent of \(12^{-15} \cdot 12^{7}\) is:
\[ \frac{1}{429981696} \]
Therefore, the final answer is:
\[ 12^{-15} \cdot 12^{7} = 12^{-8} = \frac{1}{429981696} \]