'If a+b equals b+a, why is 1/a+b different from 1/b+a?'
ab means a*b.
That's why 1/ab=1/(a*b).
But we could just as easily say 1/ab = (1/a)*b, because that distinction is only convention.
None of which excuses your horseshit belief that a(b)^2^ occasionally means (ab)^2^.