how can I increase my root available?

Can anyone tell me how to increase the space available to the root space to say 500 gigs??

check the mount point.
Probably it is in different disk or block storage

try this command to see disk info:
df -h

you might search Google for more commands and how to mount a disk

since this is a LVM you would need to check for available space with pvscan but you probably don't have any available space so you would need to decrease /home and then increase root partition, note that you can increase space online but to decrease home you would need umount it.

with a decrease, if you don't know what you are doing you can get all your data lost so do a backup of home before