Question

If `x =(4)/((sqrt(5)+1)(root4(5)+1)(root8(5)+1)(root16(5)+1))`. Then the value of `(1+x)^(48)` is-
A. 5
B. 25
C. 125
D. 625

Answer 1

Correct Answer - C

Answer 2

Correct option is C. 125

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

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

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

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

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

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

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

⇒ \(x +1 = \sqrt[16]5 - 1+ 1 = \sqrt[16]5\)

So, \((1 + x)^{48}\)

\(= (\sqrt[16]5)^{48} \)

\(= (5)^{48/16}\)

\(= 5^3\)

\(= 125\)