Dividing, multiplying, adding & subtracting algebra expressions with exponents

Anyone know why don't exponents change with addition and subtraction and the opposite with division and multiplication?

I know how to do the equations and everything, but I really want to know WHY.

My thanks goes to anyone who can help!

Edit:

If you don't know what I mean...
I was looking for an explanation on why 2x^2+2x^2 = 4x^2
whereas 2x^2 multiplied by 2x^2 = 4x^4

Can you follow the pattern?

2 apples + 2 apples = 4 apples

$\displaystyle 2x + 2x = 4x$

$\displaystyle 2x^2+2x^2 = 4x^2$

And, of course, (2 apples)(2 apples)= 4(apples x apples)= 4 apples^2!