Which one among $$\root 3 \of 6 ,\,\root 2 \of 5 $$  and $$\root 6 \of {12} $$  is the largest?

Solution (By Examveda Team)

$$\eqalign{ & \root 3 \of 6 ,\,\root 2 \of 5 {\text{ and }}\root 6 \of {12} \cr & {\text{Take LCM of }}\left( {3,\,2{\text{ and }}6} \right) = 6 \cr & {6^{\frac{{1 \times 6}}{3}}},\,{5^{\frac{{1 \times 6}}{2}}},\,{12^{\frac{{1 \times 6}}{6}}} \cr & 36,\,125{\text{ and }}12 \cr & {\text{Clearly, }}\root 2 \of 5 {\text{ is the largest}}{\text{.}} \cr} $$