To simplify the expression \((12 \cdot 12 \cdot 12) \cdot (12 \cdot 12 \cdot 12)\) using the Power Rule of Exponents, we first represent each group of \(12\)s in exponent form:
\[ 12 \cdot 12 \cdot 12 = 12^3 \]
So, we have:
\[ (12 \cdot 12 \cdot 12) \cdot (12 \cdot 12 \cdot 12) = 12^3 \cdot 12^3 \]
According to the Power Rule of Exponents, when multiplying two expressions with the same base, we add their exponents:
\[ 12^3 \cdot 12^3 = 12^{3 + 3} = 12^6 \]
Thus, the simplified equivalent expression is:
\[ \boxed{12^6} \]
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