1. **State the problem:** Simplify the expression $$\left(\frac{(2^4)^2}{(2^4)^3 \cdot 2^3}\right)^2$$. 2. **Recall the exponent rules:** - Power of a power: $$(a^m)^n = a^{m \cdot n}$$ - Product of powers with the same base: $$a^m \cdot a^n = a^{m+n}$$ - Quotient of powers with the same base: $$\frac{a^m}{a^n} = a^{m-n}$$ 3. **Simplify inside the parentheses:** - Calculate $(2^4)^2 = 2^{4 \cdot 2} = 2^8$ - Calculate $(2^4)^3 = 2^{4 \cdot 3} = 2^{12}$ So the expression inside the parentheses is: $$\frac{2^8}{2^{12} \cdot 2^3}$$ 4. **Combine the denominator powers:** $$2^{12} \cdot 2^3 = 2^{12+3} = 2^{15}$$ 5. **Rewrite the fraction:** $$\frac{2^8}{2^{15}} = 2^{8-15} = 2^{-7}$$ 6. **Apply the outer exponent 2:** $$\left(2^{-7}\right)^2 = 2^{-7 \cdot 2} = 2^{-14}$$ 7. **Final answer:** $$\boxed{2^{-14}}$$ This means the simplified form of the original expression is $2^{-14}$, which is a very small positive number equal to $\frac{1}{2^{14}}$.