4) \( (\sqrt[8]{16})^{-4} \)

First, let's simplify \( \sqrt[8]{16} \). We know that \( 16 = 2^4 \), so \( \sqrt[8]{16} = (2^4)^{1/8} = 2^{4/8} = 2^{1/2} = \sqrt{2} \). Now, we need to calculate \( (\sqrt{2})^{-4} \). Using the power of a power rule, we can rewrite this as \( \frac{1}{(\sqrt{2})^4} \). Since \( (\sqrt{2})^4 = (2^{1/2})^4 = 2^{2} = 4 \), we find that \( (\sqrt{2})^{-4} = \frac{1}{4} \). So, \( (\sqrt[8]{16})^{-4} = \frac{1}{4} \).