How could you establish the formula of #ZnSO_4*7H_2O#?

Possibly, the best means would be by complexometry.

So, perhaps you would take thesalt, #ZnSO_4*7H_2O#, or one of its several hydrates, and perform a titration with #"EDTA"#.................

And #"EDTA"# reacts with #Zn^(2+)# with 1:1 stoichiometry. With standardized #"EDTA"#, and an accurate mass of the hydrate, you could establish the degree of hydration of the zinc salt.

If you weigh an amount of hydrate and heat it (not too hot!) untill the weigh stays constant, then:

End weight represents anhydrous #ZnSO_4#, which you can convert to mols.

Weight loss represents the cristal water removed, which you also convert to mols.

The ratio of the two gives you the #x# in #ZnSO_4*xH_2O#

Note:
There are three ways this can go wrong:
(1) If you don't heat long enough, not all the water will be gone, and you'll end up with a lower number for #x#
(2) If the sample of hydrated salt is already partly dehydrated, you'll also get an #x# that is too low.
(3) If you heat too long, or too hot, part of the zinc sulphate may disintegrate: #ZnSO_4->ZnO+SO_3(g)# and you will get a value of #x# that is too high.