When you work in a corporate office or a shared space, you are bound to have conflicts with other personalities around you. You share the majority of the day with your colleagues, so it's no surprise when they get on your nerves, whether through differing schedules or annoying habits. Chewing gum is a common complaint because the noise can be aggravating, particularly if it is loud and sloppy, so attempt professional, kind ways to ask your co-worker to stop.
Invite your colleague to lunch or another more casual environment to ask her to tone down the gum chewing. Be honest but direct in explaining that a quiet and harmonious office environment is important to your ability to work. Let her know that you hope she is not offended. Perhaps your colleague was not aware the gum chewing was loud or that it affected anyone else, so mentioning it in a relaxed environment and offering it as a friendly request will make the conversation smoother and friendlier.
Offer an alternative if she needs something to chew. Provide her with a dish of hard candy or something chewy, such as Twizzlers. Presenting another option is a nice way to get her to stop chewing gum.
Use a casual reminder should the loud chewing persist. For example, "Jane, remember that conversation we had at lunch last week about the gum chewing? I wonder if there is some way we could meet in the middle on this." This approach is still friendly but not threatening or menacing and allows the offender an opportunity to remedy the situation.
Put it in writing. Send an email reminder that you are unable to work with the loud noise, asking her to stop again. This also ensures that you have a copy of your request in writing should you need to take further action.