What is the easiest way to solve with out a calculator ~ Rank the numbers
A =2^1/ 3 , B =3^1/ 4 and C =5^1/ 6 from smallest to largest
If 2^(1/3) < 3^(1/4), then
(2^(1/3))^3 < (3^(1/4))^3
2 < 3^(3/4)
2^4 < (3^(3/4))^4
16 < 3^3
16 < 27, which is true.
Therefore 2^(1/3) < 3^(1/4)
The actual proof requires you to start with 16 < 27 (which is a known true statement) and get to 2^(1/3) < 3^(1/4), but it's essentially the same, just going in reverse order.
Do the same for the other combinations of numbers you have been given.