You can see how many Gem Fragments you have in the Gem window.
In the Gem Exchange window, select the Gem you wish to exchange your Gem Fragments for in order to see how many Gem Fragments are required.
If you have enough Gem Fragments, simply tap _Exchange_ to receive your Gem. The Gem will be automatically added to your Gem bag.
The more powerful the Gem, the more Gem Fragments required.
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Game Guide for Age Of Warring Empire