Simplifying Rational Expression (Excluded Values) I would like some opinions on a problem that I am working. The problem is below.
Simplify the expression.
$$\frac{3n+6}{n+6}\cdot \frac{2n-10}{3n+6}$$
So, I simplified and got
$$\frac{2n-10}{n+6}$$
I understand why the two expressions above are not equal when n = -2. I would like to know if the two expressions are equal when n = -6. Is $$\frac{264}{0}=\frac{-22}{0}$$ a true statement? I know that the two fractions written above are undefined. Is it okay to say the the two rational expressions above that are equivalent when n = -6 since they both produce similar results (undefined = undefined)?
 A: Not unless you define what you mean by 'equivalent'. Division by $0$ traditionally has no meaning, so I guess they could be 'equivalent' in their 'meaninglessness'.
Notice that regardless of how you define 'equivalent', do not mistake 'equivalent' for 'equal'. The equality symbol has well defined meaning and properties, and division by $0$ does not interact properly with it.
A: You don't talk of two expressions being equal, by talking about if they are equal  at the points they are both undefined.  You talk of two expressions being equal if the are equal at all the points where they are defined and if they are both defined and/or undefined at the same points.
$\frac{3n+6}{n+6}\cdot \frac{2n-10}{3n+6}$ and $\frac{2n-10}{n+6}$ are not both equal at $n= -6$.  They are both undefined at $n=-6$.
To be equal they must i) both be equal at all points they are both defined.  And if $n \ne -6$ and $n\ne -2$ they are equal.  ii) be (un)defined at the same points.  And they are both undefined at $n= -6$ and they are both defined for all $n\ne -6; n\ne -2$. BUT $\frac{2n-10}{n+6}$ is defined at $n=-2$ (it is equal to $-\frac 72$) whereas $\frac{3n+6}{n+6}\cdot \frac{2n-10}{3n+6}$ is not definied at $n=-2$.  So the expressions are not equal.
....
Perhaps a clear example of what you are trying to express.
$\frac {x+1}{x^2 + 2x + 1} = \frac 1{x+1}$.
These expression are equal because if $x \ne -1$ then we have the terms are equal.
But  if $x$ does equal $-1$ we don't have the two expressions equal.  And saying $\frac 00 = \frac 10$ is FALSE.
They are not equal at $x=-1$.
But they are both undefined at $x= -1$.  And so, at all points we have either the two expressions both equal or they are both undefined.  So they are equal.
....
Anyway.... we don't have $\frac{3n+6}{n+6}\cdot \frac{2n-10}{3n+6}=\frac{2n-10}{n+6}$ but we do have $\frac{3n+6}{n+6}\cdot \frac{2n-10}{3n+6}=\frac{2n-10}{n+6}_{n\ne -2}$ where that notation, $_{n\ne -2}$ means we are restricting $\frac {2n-10}{n+6}$ only to values where $n\ne -1$.
These expressions are equal as they are both undefined and $n=-6$ and at $n=-2$ (the second is not defined ant $n=-2$ because we said it wasn't) and they are equal everywhere else.
....
P.S.  You wrote $2n+10$ rather than $2n-10$.  I'm assuming that was a typo.
