Why is N+B not equal to R+P?

It's oftentimes the case, especially in certain openings (like the Italian), that it becomes possible to exchange your knight and bishop for your opponent's rook and pawn. In the Italian opening, and similar openings, this is most often the case by taking the f7 pawn while your opponent is castled short.

In theory this should be about an equal exchange (if we use the classical piece values, it would be 6 points of material exchanged for 6 points of material). Plus, if done in the opening with the opponent's f7 pawn against a castled king, it kind of hurts the opponent's king's position and safety (no f7 pawn, no rook protecting the king).

Yet, it's traditionally considered a disadvantageous exchange. A knight+bishop is considered stronger than a rook+pawn. But why is this so? For all the years I have played chess, and known about this principle, I have never really heard a good explanation of why.

Well think about it this way. How good is the rook in the beginning of the game? In most cases it's not really active. Rook is the middlegame and endgame piece most of the time. As a contrast, minor pieces are the first to be developed. If you exchange 2 developed minor pieces in the opening phase, your opponent will have more developed pieces in the beginning of the game and thus will have more activity.

Not to mention, rook and a pawn are rarely better than 2 minor pieces, even later on. In the opening however, sometimes even rook and 2 pawns are not really enough for 2 minor pieces.

For one thing, a Bishop plus a Knight can doubly-attack a target... while a Rook plus Pawn can almost never do so.

A Bishop plus a Knight can attack twenty-one squares between the two of them, while a Rook plus a Pawn can at most attack sixteen.

Good points. The second one is something I used to do with pieces - counting the number of squares they can control. This is a good general concept to remember. 

By the way, another way to view my first post is this:

It is rarely a good idea to develop a piece and then just exchange it for a piece that is still on its starting square (for instance moving a bishop and then exchange it for an undeveloped knight for instance). In a same manner, it is not a great idea to move a knight, move a bishop and then just exchange those two pieces for a pawn that never moved and a rook that moved only due to a short castle in this case (which is actually a king move in the first place, so you might say that you traded 2 minor pieces plus 2 tempi for a rook and a pawn).

That's well put. These are the reasons why I'd generally take the two minors over the R+P.

Much also depends on how good the extra pawn is. The more advanced the pawn is, the more it's worth.

Well, let's start from the beginning and do 3 quick lessons...

1
A queen's power to attack and defend is the same as a pawn right? (all pieces capture one at a time, and can be captured in one move too). So why is the queen better? A typical beginner exercise is the coach places a piece in the center of the board and has the beginner count how many squares it can move to. A queen is worth more due to its mobility.

2
Now extend the idea of value from mobility to like pieces. In other words a centralized knight is very often better than a knight on the edge of the board. A "good" bishop (one not blocked by friendly pawns) is often better than a "bad" bishop (one that is blocked by friendly pawns.

3
Put it all together. You compare the mobility of a rook + pawn to a knight + bishop. It's worth noting too that in the first half of the game rooks are not worth much. We consider them as 5, because as the game inevitably leads to pawns coming off the board, the rooks grow into their full potential, which is about 5... but during the first 15 moves they're typically  not very good. Minor pieces (knights and bishop) are the most important in terms of activity during the first 10-15 moves. This is partly due to their lower value (pieces of higher value can be chased, but mobile pieces of lower value are more permanent).

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Bonus 4th lesson would be that mobility is just a means to an end. The ultimate goal (positionally speaking) is to come into contact with weak enemy pawns or squares around the enemy king. In the opening you place pieces in the center, and on open lines, because these conditions maximize the likelihood of coming into contact with important points later in the game.

If you didn't know these things (intellectually or intuitively) then you had very bad chess teachers

To be fair, I don't play Dragon Sicilian, so I don't have the required experience to say for sure. I will try to guess though.

Fianchetto bishop traded off, with a king castled on that side can probably be really bad if queens are on the board, probably even more so if there is a pawn storm on the way. If queen trade has happened, I would think that it can be a lot safer to grab material. If not...  I would probably have to think long about that trade. It seems risky.

It probably depends on the stage of the game. I think the reason a knight and bishop is not equal to a rook and pawn is because even though both have a combined value of about 6, the rook is more powerful and the pawn can promote. 

Late in the game a knight and bishop will draw or lose. While a rook and pawn will likely draw or win. It would be a rare situation where the bishop and knight can force a win over the rook and pawn. But the vast majority of the time they will probably draw. 

"It probably depends on the stage of the game"

That's a very good point! In the beginning of the game, minor pieces can come into play quickly and make an immediate impact. Rooks are stuck in the corners and take a longer time to come into play. And the rooks need open files to show their strength.  Towards the end of the game, when the lines are wide open, a rook and pawn can often defeat two minor pieces. In the earlier stages, the two , minors are almost always better

     Exactly. In the Italian game, white has set out to threaten black's king. Using up the two closest attacking pieces just dissipates the attack and neutralizes the whole plan.

     In the ending, R+P can often force the B or N to sacrifice itself for the soon-to-queen pawn, leaving the R with excellent winning chances and no way of losing.

So is exchanging N+B for R+P advantageous in the endgame?

Depends entirely on the position. In many positions, the minor pieces are better.

You do not only have to look at material. If you play e4, Bc4, Nf3, Ng5, Nxf7, Bxf7+ and black plays e5, Nf6, Bc5, O-O, Rxf7, Kxf7 then white is left with 0 developed piece into play and black with 2 developed pieces into play. White has lost 2 tempi.