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Which among $2^{\frac{1}{2}} , 3^{\frac{1}{3}}, 4^{\frac{1}{4}}, 6^{\frac{1}{6}} \text{ and } 12^{\frac{1}{12}}$ is the largest?

1. $2^{\frac{1}{2}}$
2. $3^{\frac{1}{3}}$
3. $4^{\frac{1}{4}}$
4. $6^{\frac{1}{6}}$
5. $12^{\frac{1}{12}}$

Here the bases of numbers can't be made same so making their powers equal

1. $2^{\frac{1}{2}} = 2^{({6})\frac{1}{12}} = 64^{\frac{1}{12}}$

2. $3^{\frac{1}{3}} = 3^{({4})\frac{1}{12}} = 81^{\frac{1}{12}}$

3. $4^{\frac{1}{4}} = 4^{({3})\frac{1}{12}} = 64^{\frac{1}{12}}$

4. $6^{\frac{1}{6}} = 6^{({2})\frac{1}{12}} = 36^{\frac{1}{12}}$

5. $12^{\frac{1}{12}}$

Here clearly we can see  $3^{\frac{1}{3}} = 3^{({4})\frac{1}{12}} = 81^{\frac{1}{12}}$ is the Largest value.

Hence (B) $3^{\frac{1}{3}}$ is the Answer.

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