My grandmother is in the early stages of dementia. She can be mean and say cruel things. How do I deal with someone exasperating?

can you talk to your mother? She needs to understand that your grandmother is changing, that she is not completely in control of what she says and your mother needs to not take it personally. Also, if she is very angry at your grandmother it might help to explain that bay caring for your grandmother she is really helping you, and being a good mother to you, and a good grandmother to her grandchildren.
I have found that reframing the issue can really help in enlisting assistance from relatives. Emphasize that that your grandmother has changed, and that you, not just your grandmother need the help.
Also, at least initially spell things out for your mom, I had my mother start doing basic house cleaning chores and shopping, that way she could take a load off me even when it was difficult for her to be around my grandmother (like your grandmother, mine can be hard on the people around her.)
Finally, the only way I have found to deal with my grandmothers negativity and criticism sounds a lot like yours, I give a noncommittal, nonjudgmental reply and change the subject. And when she is being really bitter, I make sure she is as safe and comfortable as possible and I make myself busy in another part of the house. I am having to learn that I cannot always make everything right for her.

YOU NEED TO GET MEDS. Yep please have an psychaitrist see your grandma. You need to be there as well to fill in the parts she does not answer like how her actions and her personality is. They like to act all nice and sane in public and then with family the viper comes out. Some meds help with the personality changes and mood swings and or depression they start to get.