BurritoRoot may not work on the 6.0 firmware, please update to a newer firmware if it doesn't work for you.
BurritoRoot is an application that allows the android debug bridge (adb) to be ran as root simply by running the command 'adb root'.
As of today, it is the only software based way of gaining root on the Kindle Fire. It also happens to be 100% working and reliable. Unlike other solutions it does not require modified hardware/cable, or exploiting a vulnerability that can cause instability (and doesn't work on current firmware).
I will keep this updated with a current link to BurritoRoot as I update it, and BurritoMOD when I finish it.
Burrito Root: http://www.androidpolice.com/2011/12/23/exclusive-how-to-easily-root-the-amazon-kindle-regardless-of-its-software-version/
Currently supporting all known firmware but 6.0.
After running BurritoRoot, and clicking 'Root', adb can be forced to run as root by running the command 'adb root'. It is that easy.
The unroot option only disables the 'adb root' command. It DOES NOT reverse changes you have made to your system.
I am actively seeking to bring other variations of BurritoRoot to more tablets and phones. My track records is fairly good, but lack of hardware is keeping me down. If my solution has helped you, and you wish to help me spread the Burrito, consider buying one of my apps. If non intrest you, consider buying the donation app that does nothing (will serve as a license key for BurritoMOD when it is released). https://market.android.com/developer?pub=Cunning+Logic
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