1. **State the problem:** Simplify the expression $2 (2\sqrt{4})^{-2}$. 2. **Recall the rules:** - The square root of 4 is $\sqrt{4} = 2$. - Negative exponents mean reciprocal: $a^{-n} = \frac{1}{a^n}$. - Multiplication and exponentiation follow order of operations. 3. **Simplify inside the parentheses:** $$2\sqrt{4} = 2 \times 2 = 4$$ 4. **Rewrite the expression:** $$2 (4)^{-2}$$ 5. **Apply the negative exponent:** $$2 \times \frac{1}{4^2} = 2 \times \frac{1}{16}$$ 6. **Multiply:** $$\frac{2}{16}$$ 7. **Simplify the fraction by canceling common factors:** $$\frac{\cancel{2}^1}{\cancel{16}^8} = \frac{1}{8}$$ **Final answer:** $$\boxed{\frac{1}{8}}$$