Bamboo is a grass. You can chop it down, but it will sprout back
out. You either have to dig up the roots, also, or kill the plant
with a herbicide. Check with the nearest forest service, as they have
a potent enough tree killer that will do the job and only they are
allowed to use that herbicide. I doubt you can do that big of job,
yourself.

There is a highly potent tree killer, pretty expensive, for
consumers. I don't know the name, right off, but I can find out by
tomorrow, probably. You spray it about 18" above the ground and it
kills the plant. You might have trouble spraying the bamboo in the
interior of that big of patch, if it's so dense you can't navigate
within.

Once a herbcide is used and the plants die, you will still have the
task of removing the dead bamboo.

The roots are as deep as 1'. You'll have lots of digging to remove
all those roots, if you try to remove everything manually, while it's
still alive. If a herbicide is used to kill the above ground plant,
the roots can remain. They'll eventually decay. Tilling the dead
roots will speed their decaying.

Sonny