Is it obligatory to make up the days during which toothpaste was used in the daytime in Ramadan due to ignorance of what invalidates the fast, and is expiation required along with making up those days?

Using toothpaste during the daytime in Ramadan (definition) is permissible, as long as nothing reaches the throat intentionally. If something reaches it unintentionally, the fast is not invalidated. However, swallowing it intentionally invalidates the fast, unless it was done out of ignorance; in which case, the fast is not invalidated, and no expiation is required. Therefore, there is no need for a make-up fast (qada') or expiation (kaffarah (definition)) for past days in which you swallowed toothpaste out of ignorance. However, in the future, you must be careful not to swallow it, and it is better to postpone its use until after breaking the fast.