Question : The value of $\sqrt{2\sqrt[3]{4\sqrt{2\sqrt[3]{4\sqrt{2\sqrt[3]{4.......}}}}}}$ is:
Option 1: 2
Option 2: $2^{2}$
Option 3: $2^{3}$
Option 4: $2^{5}$
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Correct Answer: 2
Solution : Let $x=\sqrt{2\sqrt[3]{4\sqrt{2\sqrt[3]{4\sqrt{2\sqrt[3]{4.......}}}}}}$ By squaring both sides, we get, $⇒x^{2} =2×\sqrt[3]{{4×x}}$ By cubing both sides, we get: $⇒x^{6}=8×4x$ $⇒x^{5}=32=2^{5}$ $\therefore x=2$ Hence, the correct answer is 2.
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