Previously the Amazon Prime Video app only worked on Android phones, but Amazon has been expanding compatibility and now Android tablets will work as well. Recently Amazon expanded playback to the Fire tablet as well as iOS devices, but Android tablet users were still not able to play videos on their devices.
To browse content users need to launch Prime Instant Video from their devices web browser, and they will then be able to watch on their tablets. The app is only available through Amazon's own Appstore at this point, which you can install here if you need.
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