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What would I do without your smart mouth
This means that she is an intelligent woman. She might be well educated. You don't call your love ones' smart mouth if it means something else. Even if she is rude or she doesn't have her day and is a little bit grumpy, he will still love her as much. Probably that she has makes smart remarks, and a sort of attitude, and he likes that about her. She is so special to him that it is unimaginable for him to imagine a world without her. The way she talks, the gestures she makes are addictive.