as we see that the roots are doubleing in term to term so multiply numeraator and denominator by $(\sqrt[16]{5}-1)$ we get
$(\frac{4}{(\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[8]{5}+1)(\sqrt[16]{5}+1)} +1)^{48}$
= $(\frac{4*(\sqrt[16]{5}-1)}{(\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[8]{5}+1)(\sqrt[16]{5}+1)(\sqrt[16]{5}-1)} +1)^{48}$
= $(\frac{4*(\sqrt[16]{5}-1)}{(\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[8]{5}+1)(\sqrt[8]{5}-1)} +1)^{48}$ using $a^2-b^2$ formula for last 2 terms
= $(\frac{4*(\sqrt[16]{5}-1)}{(\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[4]{5}-1)} +1)^{48}$ using $a^2-b^2$ formula for last 2 terms-
= $(\frac{4*(\sqrt[16]{5}-1)}{(\sqrt{5}+1)(\sqrt{5}-1)} +1)^{48}$ using $a^2-b^2$ formula for last 2 terms-
= $(\frac{(4*\sqrt[16]{5}-1)}{4}+1)^{48}$
= $(\sqrt[16]{5})^{48} = 5^3 = 125$
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