The greatest one of $$\sqrt 2 ,$$  $$\root 3 \of 3 ,$$  $$\root 6 \of 6 ,$$  $$\root 5 \of 5 $$  is = ?

$$\eqalign{ & \sqrt 2 \to {2^{\frac{1}{2}}} \to {2^{\frac{{15}}{{30}}}} = \root {30} \of {{2^{15}}} = \root {30} \of {32768} \cr & \root 3 \of 3 \to {3^{\frac{1}{3}}} \to {3^{\frac{{10}}{{30}}}} = \root {30} \of {{3^{10}}} = \root {30} \of {59049} \cr & \root 6 \of 6 \to {6^{\frac{1}{6}}} \to {6^{\frac{5}{{30}}}} = \root {30} \of {{6^5}} = \root {30} \of {7776} \cr & \root 5 \of 5 \to {5^{\frac{1}{5}}} \to {5^{\frac{6}{{30}}}} = \root {30} \of {{5^6}} = \root {30} \of {15625} \cr & {\text{So }}\root 3 \of 3 {\text{ is the greatest}}{\text{.}} \cr} $$