From the given position, is it better to take with rook or the a-pawn and why ?
Black to move:
The engine suggested the move
There must be some strategical or tactical point (ignore the arrows in the diagram).
After axb4 the a2-pawn is a fixed weakness that can easily be further attacked with Ra8. Defending this pawn will force whites heavy pieces into passive positions.
Taking with the queen or the rook on b4 will allow Rb1 with a rook exchange. Even if this rook exchange happens on b4 and therefore creates the same pawn structure as after an immediate axb4, the defence of a weak a-pawn will be much easier without the rooks. In fact white might be able to exchange the a-pawn for the b-pawn relatively quickly.
In my opinion, capturing with the pawn is the correct move, and the logic behind the move is twofold and more of a strategic one.
First, you remove a weakness of your own. Black's
a pawn is an isolated one and it creates a pawn island of its own. Therefore, after capturing with the pawn, you no longer have any isolated pawns and you've only got just one pawn island, instead of two (typically a pawn structure with less pawn islands is superior to a pawn structure one with more pawn islands).
Also, capturing with the pawn exposes a weakness in the White camp, namely the
a pawn. It is also an isolated pawn and by capturing with the
b pawn Black opens a file against it (creates a target) and can later in the game exploit the weakness by adding pressure along the opened
1...axb4 does two things:
Firstly, you can place pressure open file with an eventual Ra8. He can never defend the pawn actively because you would have the pin using move b3. The only defense would be to cram his Rook into a corner with the ugly Qb2 followed by Ra1 (by white)
Secondly, you are saving a pawn island (which was the a pawn) and moving it into the rest of the group. It can now be much better supported by a move like c5 sometime
I would take the rook with the pawn, because it allows you to move the rook to the corner and then pressure the white queen with your queen, in other words Ra8 (black does whatever) Qa2