The Kindle app on Fire tablets is based on the Kindle for Android app. The exact path where it places Kindle book files depends on the model of Fire tablet and whether or not it has an SD card.
When you locate the correct path it will have as part of the path: .../Android/data/com.amazon.kindle/files/Books
That should have a subfolder based on your Amazon account ID, such as: amzn1.account.XXXXXXXXXXXXXXXXXXXXXXXXXXXX
Under that there will be one folder for each book on the device named by the 10 character ASIN of the book. For example: B082S1N87S or B003RCJNXI
Under each of those folders will be the files that make up that book along with some supplemental data. These days most books will be in KFX format with file names like CR!7TD9PMT74D3EN3ZWG2HDXDYY5T7C.kfx and amzn1.drm-voucher.v1.e4701cd1-97fb-42c2-b356-b1f7c78eeb54.ast. Books in older Kindle formats, such as MOBI or AZW3/KF8, will have a file with the extension of .prc.
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