1. State the problem: Solve the exponential equation $$12^{x-2} = 3^{3x} \cdot 2^{6x}$$ for $x$. 2. Express all terms with prime factors: Note that $12 = 2^2 \cdot 3$, so $$12^{x-2} = (2^2 \cdot 3)^{x-2} = 2^{2(x-2)} \cdot 3^{x-2}.$$ Thus, the equation becomes $$2^{2(x-2)} \cdot 3^{x-2} = 3^{3x} \cdot 2^{6x}.$$ 3. Group the terms by base: Collecting powers of 2 and 3 on each side gives $$2^{2x-4} \cdot 3^{x-2} = 3^{3x} \cdot 2^{6x}.$$ 4. Equate powers of corresponding bases: Since bases 2 and 3 are prime and appear on both sides, equate exponents: - For base 2: $$2x - 4 = 6x,$$ - For base 3: $$x - 2 = 3x.$$ 5. Solve the system: - From base 2: $$2x - 4 = 6x \Rightarrow -4 = 4x \Rightarrow x = -1.$$ - From base 3: $$x - 2 = 3x \Rightarrow -2 = 2x \Rightarrow x = -1.$$ 6. Both equations agree on $x = -1$, which is the solution. 7. Final answer: $$\boxed{x = -1}.$$