Which of the following relation is/are true?
$$\eqalign{ & {\text{I}}.{\left( {27} \right)^{\frac{1}{3}}} > {\left( {13} \right)^{\frac{1}{2}}} < {\left( {47} \right)^{\frac{1}{6}}} \cr & {\text{II}}.{\left( {23} \right)^{\frac{1}{3}}} < {\left( {49} \right)^{\frac{1}{2}}} < {\left( {52} \right)^{\frac{1}{6}}} \cr & {\text{III}}.{\left( {53} \right)^{\frac{1}{6}}} < {\left( {41} \right)^{\frac{1}{3}}} < {\left( {37} \right)^{\frac{1}{2}}} \cr} $$

Solution (By Examveda Team)

$$\eqalign{ & {\text{Take LCM of }}\left( {2,\,3\,\& \,6} \right) = 6 \cr & {\text{I}}.{\left( {27} \right)^{\frac{1}{3}}} > {\left( {13} \right)^{\frac{1}{2}}} < {\left( {47} \right)^{\frac{1}{6}}} \cr & = {\left( {27} \right)^{\frac{2}{{3 \times 2}}}} > {\left( {13} \right)^{\frac{3}{{2 \times 3}}}} < {\left( {47} \right)^{\frac{{1 \times 1}}{{6 \times 1}}}} \cr & = {\left( {27} \right)^{\frac{2}{6}}} > {\left( {13} \right)^{\frac{3}{6}}} < {\left( {47} \right)^{\frac{1}{6}}} \cr & = {\left( {729} \right)^{\frac{1}{6}}} > {\left( {2197} \right)^{\frac{1}{6}}} < {\left( {47} \right)^{\frac{1}{6}}} \cr & {\text{This statement is false}} \cr & {\text{II}}.{\left( {23} \right)^{\frac{1}{3}}} < {\left( {49} \right)^{\frac{1}{2}}} < {\left( {52} \right)^{\frac{1}{6}}} \cr & = {\left( {23} \right)^{\frac{2}{{3 \times 2}}}} < {\left( {49} \right)^{\frac{3}{{2 \times 3}}}} < {\left( {52} \right)^{\frac{{1 \times 1}}{{6 \times 1}}}} \cr & = {\left( {529} \right)^{\frac{1}{6}}} < {\left( {117649} \right)^{\frac{1}{6}}} < {\left( {52} \right)^{\frac{1}{6}}} \cr & {\text{This statement is false}} \cr & {\text{III}}.{\left( {53} \right)^{\frac{1}{6}}} < {\left( {41} \right)^{\frac{1}{3}}} < {\left( {37} \right)^{\frac{1}{2}}} \cr & = {\left( {53} \right)^{\frac{1}{{6 \times 1}}}} < {\left( {41} \right)^{\frac{2}{{3 \times 2}}}} < {\left( {37} \right)^{\frac{3}{{2 \times 3}}}} \cr & = {\left( {53} \right)^{\frac{1}{6}}} < {\left( {1681} \right)^{\frac{1}{6}}} < {\left( {50653} \right)^{\frac{1}{6}}} \cr & {\text{This statement is true}} \cr} $$