If $$x = \sqrt {a\root 3 \of {b\sqrt {a\root 3 \of b } } } \,.....\,\infty ,$$     then the value of x is:

Solution (By Examveda Team)

$$\eqalign{ & {\text{Given}} \cr & x = \sqrt {a\root 3 \of {b\sqrt {a\root 3 \of b } } } \,.....\,\infty \,.\,.\,.\,.\,.\,.\,\left( {\text{i}} \right) \cr & {\text{On squaring both sides}} \cr & \Rightarrow {x^2} = a\root 3 \of {b\sqrt {a\root 3 \of b } } \,.....\,\infty \cr & {\text{On cubing both sides}} \cr & \Rightarrow {x^6} = {a^3}b\,\sqrt {a\root 3 \of {b\sqrt {a\root 3 \of b } } } \,.....\,\infty \cr & \Rightarrow {x^6} = {a^3}b\,x{\text{ from equation }}\left( {\text{i}} \right) \cr & {\text{On dividing above equation by }}x{\text{ we get}} \cr & \Rightarrow \frac{{{x^6}}}{x} = \frac{{{a^3}bx}}{x} \cr & \Rightarrow {x^5} = {a^3}b \cr & \Rightarrow x = \root 5 \of {{a^3}b} \cr} $$