The value of $$\sqrt{2\sqrt[3]{4}\sqrt{2\sqrt[3]{4}}\sqrt[4]{2\sqrt[3]{4}}.....}$$ is

To find : $$y=\sqrt{2\sqrt[3]{4}\sqrt{2\sqrt[3]{4}}\sqrt[4]{2\sqrt[3]{4}}.....}$$

Let $$2\sqrt[3]4=x$$

=> $$y=\sqrt{(x)\times(\sqrt{x})\times(\sqrt[4]{x})\times.......}$$

=> $$y^2=(x)^{[1+\frac{1}{2}+\frac{1}{4}+......+\infty]}$$

Now, sum of infinite G.P. = $$\frac{a}{(1-r)}$$, where first term = $$a=1$$ and common ratio = $$r=\frac{1}{2}$$

=> $$y^2=(x)^{\frac{1}{1-\frac{1}{2}}}$$

=> $$y^2=(x)^2$$

=> $$y=x$$

$$\therefore$$ $$\sqrt{2\sqrt[3]{4}\sqrt{2\sqrt[3]{4}}\sqrt[4]{2\sqrt[3]{4}}.....}=2\sqrt[3]4$$

=> Ans - (A)