1. Stating the problem: Simplify the expression $$\frac{2 \cdot (2^2)^3}{(2^2)^3}$$. 2. Use the exponentiation rule: $$(a^m)^n = a^{m \cdot n}$$. 3. Simplify the powers: $$(2^2)^3 = 2^{2 \cdot 3} = 2^6$$. 4. Substitute back: $$\frac{2 \cdot 2^6}{2^6}$$. 5. Multiply numerator: $$2 \cdot 2^6 = 2^1 \cdot 2^6 = 2^{1+6} = 2^7$$. 6. Expression becomes: $$\frac{2^7}{2^6}$$. 7. Apply division rule for exponents: $$\frac{a^m}{a^n} = a^{m-n}$$. 8. Simplify: $$2^{7-6} = 2^1 = 2$$. Final answer: $$2$$.