Question

If x = 4/(√5+1) (4√5 +1) (8√5 +1 ) (16√5 +1). Then the value of (1+x)^48 is. It is 4th root of 5 and 8th root of 5....

Answer 1

Answer:

125

Step-by-step explanation:

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

$\implies x=\frac{4(\sqrt[16]{5}-1)}{(\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[8]{5}+1)(\sqrt[16]{5}+1)(\sqrt[16]{5}-1)}$

$\implies x=\frac{4(\sqrt[16]{5}-1)}{(\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[8]{5}+1)(\sqrt[8]{5}-1)}$

$\implies x=\frac{4(\sqrt[16]{5}-1)}{(\sqrt{5}+1)(\sqrt[4]{5}+1)(\sqrt[4]{5}-1)}$

$\implies x=\frac{4(\sqrt[16]{5}-1)}{(\sqrt{5}+1)(\sqrt{5}-1)}$

$\implies x=\frac{4(\sqrt[16]{5}-1)}{5-1}$

$\implies x=(\sqrt[16]{5}-1)$

$\implies x+1=\sqrt[16]{5}$

$\therefore (x+1)^{48}=5^{\frac{48}{16}}=5^3=125$