To find the numerical equivalent of \( 12^{-5} \times 12^{7} \) using the product rule of integer exponents, we first need to remember what the product rule states. The product rule states that when you multiply two expressions with the same base, you can add their exponents. This can be expressed as:
\[ a^m \times a^n = a^{m+n} \]
In your case, the base \( a \) is \( 12 \), and you have exponents \( -5 \) and \( 7 \).
Applying the product rule:
\[ 12^{-5} \times 12^{7} = 12^{-5 + 7} \]
Now, we calculate the exponent:
\[ -5 + 7 = 2 \]
So we have:
\[ 12^{-5} \times 12^{7} = 12^{2} \]
Next, we need to compute \( 12^2 \):
\[ 12^2 = 144 \]
Thus, the numerical equivalent of \( 12^{-5} \times 12^{7} \) is:
\[ \boxed{144} \]
Summary:
- We identified the common base (12) and applied the product rule \( a^m \times a^n = a^{m+n} \).
- We added the exponents \( -5 + 7 \) to get \( 2 \).
- We calculated \( 12^{2} \) which equals \( 144 \).
- Therefore, \( 12^{-5} \times 12^{7} = 144 \).