If x = a 3 b a 3 b \,.....\, , then the value of x is:

Answer: Option D

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