Mormons don't "simply ask" people to pray about it, but to read the book, to make knowing whether it is true or not the focus of their desires, and then to pray and ask God, and "ponder it in their hearts", and if he has "a sincere heart, with real intent, having faith in Christ" (Moroni 10:3-5), he will know whether or not it is true.
Simply, because historical and archaeological sources are the wisdom of man, but the revelations of God can penetrate the heart. Testimonies of the Book of Mormon come just as Paul, in Galatians 1:11-12, received a testimony of the gospel and learned its truths:
11 But I certify you, brethren, that the gospel which was preached
of me is not after man.
12 For I neither received it of man, neither was I taught it, but by the revelation of Jesus Christ.
It is by the Spirit of God that one gains a true testimony. This can happen in any number of ways, according to God's will and the person's humility. Mormons don't really limit how God will answer another person's prayers. But almost universally, a feeling of peace will enter into their heart and mind.
As an aside, I think I understand why you included "prophetical" in your list along with "historical" and "archaeological" fields, in that you probably know that Mormons tend to discourage looking at Bible references for proof that the Book of Mormon is true. I think that is the difference. When one looks for "proof," their heart is not in the right place. When one seeks a divine witness, however, they are more ready for the sacred revelations of God.
Of course, this is not to say that scriptures should not be used as a reference. That would be foolish. Use the scriptures, study them, and use them to reinforce your own understanding, and enlighten your mind, and avoid deception.
Here's an official page describing how to know the Book of Mormon is true (scroll down).
Again, the best thing one can do to know whether it's true or not is to read the book with a sincere heart, with real intent, having faith in Christ. It is not merely a matter of praying about it.