Step 1:
Download & Install TowelRoot
The process couldn't be easier—start by
making sure you have installation from
"Unknown sources " enabled, then just grab the
TowelRoot apk from towelroot.com and install.
We're rooting using a pretty genius method. It
basically exploits the kernel, which freezes
Android, and while the OS is sitting there
panicking, it asks for root privileges and
Android gives them to it. Then, it copies over
the necessary root files and reboots the
phone. But because of the way this exploit
functions, you'll see a nice scary warning
when installing TowelRoot—check that you
understand the risks, then hit Install anyway.
Step 2:
Run TowelRoot
Now hit the make it ra1n button, and let the
app do its thing. It'll automatically reboot
your device, and then you'll be rooted!
Step 3:
Install SuperSU
While TowelRoot will root your device, it will
not install a root manager, which is critical for
keeping malicious apps from gaining root
access. Far and away the best root manager
is SuperSU from developer Chainfire. Head to
the Play Store to grab the app directly.

gs4.wonderhowto.com/how-to/root-any-samsung-galaxy-s4-one-click-0155622/

SuperSU chain fire