If I want to load a value from a memory which base address is at $a0 and off set $t2, why can't I do the following:
lw $s2, $a1($t2)
so what is the equivalent of the expression above?
Answer
You can't do that because there's no MIPS instruction encoding that supports such a thing. You need to do the addition yourself:
add $a2, $a1, $t2
lw $s2, 0($a2)
The lw instruction encoding looks like this:
1000 11ss ssst tttt iiii iiii iiii iiii
Where sssss is the source register number, ttttt is the destination register number, and iiiiiiiiiiiiiiii is the immediate. There's no room in that encoding (and no alternate instruction encodings) that use two registers to generate the memory address. The specific machine instruction that would get encoded from the example above is:
1000 1100 1101 0010 0000 0000 0000 0000
Since the immediate is 0, $s2 is register 18 and $a2 is register 6.