Rooting allows you to install those applications on your Galaxy Tab 3 8.0 that require super user access. There are many ways of rooting it but the best one that I personally used to perform root on my Tab 3 8.0 is this one. It makes use of using TWRP to perform root. Follow these instructions carefully. Keep in mind that your warranty will be voided after performing root.
1. Download ODIN 3.07. http://d-h.st/rNl
a. Extract to C: root for easy navigation
2. Download Recovery TWRP.
a. I used release 3 from http://d-h.st/bQh which was TWRP v22.214.171.124
b. Place this in the directory from 1a
3. Power off your Galaxy Tab 3 8.0
4 With the tablet off you will now restart in "Download Mode" by holding the "HOME" button, "Volume Down" and "Power" all at the same time for about 3-5 seconds till you see the Warning!! screen
5. Toggle the "Volume Up" to enter download mode.
6. Start the ODIN.exe program you downloaded from step 1.
7. Connect your tablet to your PC
a. Drivers were automatically downloaded and installed on my Windows 7 PC (took a minute or two)
b. Make sure you have you table added in ODIN by looking at the ID:COM or in the Message window it will say Added!!
8. In ODIN Select “PDA” by putting a check in the box and then navigate to the .tar file from step 2
9. Now select “START”
10. In a few minutes or less your tablet will reboot automatically
11. Power off your tablet again and access the newly installed recovery
a. To access recovery press and hold volume up and power (release the power button once the Samsung Galaxy Tab 3 splash screen is visible continuing to hold the volume up)
b. Hold volume up until you see the standard teamwin recovery screen
12. Once in Team Win Recovery select Reboot from the menu on the bottom right if you are holding your tablet upright like a portrait. Select “System”
13. Hopefully you will be prompted to install ROOT like I was
a. If so, choose that you do and follow the rest of the on screen instruction upon boot
b. I chose the TWRP install method instead of the paly store and my tablet reboot after downloading a few files
c. I verified with root checker and I was successfully rooted