It would be a waste of time.
Purifying gold and silver is a laborious, messy process that requires nasty chemicals. There are oddball gold sources (such as some electronic components) where this is the only way of getting sellable gold, but you need a LOT of these components to get any reasonable amount of gold from them. Some of these components even have to be crushed in order to expose the gold to chemical action.
With jewelry and coins, it's a VERY simple process of ascertaining how much actual gold is in the piece, so the gold/silver doesn't have to be extracted.
Say you had 10 14K rings weighing 1.7 oz.
Scenario 1:
You take your rings and go running down to the dealer with them. The dealer would test them, see that they're 14K, weigh them up and he'll see that they contain an ounce of pure gold.
The dealer will then pay you for an ounce of gold minus his transaction fee.
Scenario 2:
You take your rings, go buy the chemicals and the equipment to extract the gold in pure form. After about 3 days of making a mess, measuring mixtures, inhaling fumes and burning your fingers, you end up with a 1 oz slug of pure gold and go running down to the dealer with it. The dealer checks out your slug and sees that it's an ounce of pure gold.
The dealer will then pay you for an ounce of gold minus his transaction fee.
It's not worth the effort.