To simplify the expression \((12 \cdot 12 \cdot 12) \cdot (12 \cdot 12 \cdot 12)\) using the Power Rule of Exponents, we first recognize that each group of \(12\)s can be expressed as an exponent.
Each group \(12 \cdot 12 \cdot 12\) consists of three \(12\)s, which can be represented as \(12^3\).
Thus, the original expression can be rewritten as:
\[ (12^3) \cdot (12^3) \]
According to the Power Rule of Exponents, when multiplying two expressions with the same base, you add their exponents:
\[ 12^3 \cdot 12^3 = 12^{3+3} = 12^6 \]
So, the simplified equivalent expression is:
\[ \boxed{12^6} \]