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The Negative Exponent Rule states that \( a^{-n} = \frac{1}{a^n} \), where \( a \) is a non-zero number and \( n \) is a positive integer.
To apply this rule to the expression \( 3^{-12} \):
\[
3^{-12} = \frac{1}{3^{12}}
\]
So, the equivalent expression for \( 3^{-12} \) is:
\[
\frac{1}{3^{12}}
\]