does 3^1/2 and 2^1/3 belongs to Z

if you are taking about 'Sets', where 'Z' denotes an integer value ,

3^1/2 = √3  which is not an integer , so it doesn't belong to 'Z'

similarly 2^1/3 = 3√2 which is also not an integer , so it doesn't belong to 'Z'

an integer is a value that ranges from -∞ to ∞ such that they are purely numbers.

No, none of these belong to integers but both of them belongs to real numbers ie. R