How could I simplify
$\displaystyle \frac{(n+b)}{w+(n+b)}\cdot\frac{b}{w+b}
$
maybe the following might help:
try using a substitution: a = (n + b),
so we have,
$\displaystyle \frac{a}{(w+a)}$.$\displaystyle \frac{b}{(w+b)}$
multiply out the denominator and then re-substitute for 'a' and simplify
i ended up with $\displaystyle \frac{b(n+b)}{(w+b)((w+b)^2 + n)}$
perhaps someone could verify this ....
Perhaps if we rewrite, it will become apparent that this thing is factored as nicely as possible.
$\displaystyle \frac{(n+b)}{w+(n+b)}\cdot\frac{b}{w+b}=\frac{b(n+ b)}{(w+b)(w+n+b)}$
Note that
1. Nothing cancels
2. there is no way to factor any more completely.
Therefore, this quotient is in its simplest form.
If you apply those two rules, you'll never have these questions again.
$\displaystyle (n+b) / w + (n+b) x b / w + b$
Notice above that $\displaystyle (n+b)$ can cancel out leaving us with $\displaystyle 1 / w$ Now, multiply that by $\displaystyle b / w+b$, and I think it becomes $\displaystyle b / w^2 + wb$. Then, b cancels out and I think we're left with $\displaystyle 1 / w^2 + w$