\(\sqrt{\frac{8^2 \times 4^{n + 1}}{2^{2n} \times 16}}\) = \(\sqrt{\frac{2^{3(2)} \times 2^{2(n + 1)}}{2^{2n} \times 2^4}}\) = \(\sqrt{\frac{2^6 \times 2^{2n + 2)}}{2^{2n} + 4}}\) = \(\sqrt{\frac{2^6 + 2^{2n + 2)}}{2^{2n} + 4}}\) = \(\sqrt{\frac{2^{2n + 8}}{2^{2n} + 4}}\) = \(\sqrt{2^{2n + 8} \div 2^{2n} + 4}\) = \(\sqrt{2^{2n - 2n} + 8 - 4}\) = \(\sqrt{2^4}\) = \(\sqrt{16}\) = 4