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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