1989 Toyota Corolla Removing Axle

Nope, they "pop" out. The best tool to use is a slide hammer with an CV axle attachment.
You can get a long heavy screwdriver and put on slot on tripod side, (from opposite side) use hammer and whack it a few times.

Some cars have axles that are bolted to trans, but most use a "C" clip at end of axle that compresses as it is going in then expands into slot in trans.
They can sometimes be a pain to get out.

Go to autoparts store and look at replacement axle, and you will see the small "C" clip that holds axle in.