Question:
The value of $\sqrt[4]{\sqrt[3]{2^{2}}}$ is
(a) $2^{\frac{-1}{6}}$
(b) $2^{-6}$
(c) $2^{\frac{1}{6}}$
(d) $2^{6}$
Solution:
$\sqrt[4]{\sqrt[3]{2^{2}}}=\left[\left(2^{2}\right)^{\frac{1}{3}}\right]^{\frac{1}{4}}$
$=\left[2^{\frac{2}{3}}\right]^{\frac{1}{4}}$
$=2^{\frac{2}{12}}$
$=2^{\frac{1}{6}}$
$\therefore$ The value of $\sqrt[4]{\sqrt[3]{2^{2}}}$ is $2^{\frac{1}{6}}$.
Hence, the correct option is (c).
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