Here’s a guide for how to Root your Galaxy Tab 3 10.1 GT-P5210. The original xda guide is available here. You can find all the links in that post.
1. Get Odin3.07 and all the other files.
- Root.zip [http://www.mediafire.com/download/erw5iptea0awmws/Restl3ssroot.zip] Place this on your external sdcard
- TWRP Recovery [http://forum.xda-developers.com/showthread.php?t=2442915]
2. Boot Your Galaxy Tab into Download mode
Do this by shuting down, then hold the Power and Volume down Button.
Press Volume up to contiue
3. Connect it to your pc using micro-usb and start Odin, make sure that Kies is not running in the Background.
4. Flash the Recovery File
4.1. Click on PDA ans select the Recovery Package (it is the .md5 file inside the zip) 4.2. Uncheck Auto Reboot 4.3 Flash. Reboot your device when the Message board stops showing new messages (About 2 – 6 min)
5. Boot into recovery and flash root.zip
To boot into recovery hold the Power and Volume up Button. Use the Power button to select the zip file you placed on your sdcard.
Congratulations and welcome aboard. You are now rooted.