An algebra problem by Saurav Pal

Algebra Level 3

If

\[x=\frac{4}{\big(\sqrt{5}+1\big)\big(\sqrt[4]{5}+1\big)\big(\sqrt[8]{5}+1\big)\big(\sqrt[16]{5}+1\big)},\]

then what is the value of \((1+x)^{48}?\)