To simplify the expression
[tex]\[
\frac{1}{m-n}-\frac{m+n}{m^2-n^2}
\][/tex]
let's start by rewriting each term and understanding the common denominators.
First, recall that the denominator [tex]\(m^2 - n^2\)[/tex] can be factored using the difference of squares:
[tex]\[
m^2 - n^2 = (m-n)(m+n)
\][/tex]
So the original expression is:
[tex]\[
\frac{1}{m-n} - \frac{m+n}{m^2 - n^2}
\][/tex]
We substitute the factored form of [tex]\(m^2 - n^2\)[/tex]:
[tex]\[
\frac{1}{m-n} - \frac{m+n}{(m-n)(m+n)}
\][/tex]
The first fraction is already simplified, so we just rewrite it as is:
[tex]\[
\frac{1}{m-n}
\][/tex]
For the second fraction, we see that the [tex]\(m+n\)[/tex] in the numerator and the denominator cancel each other out:
[tex]\[
\frac{m+n}{(m-n)(m+n)} = \frac{1}{m-n}
\][/tex]
So the expression becomes:
[tex]\[
\frac{1}{m-n} - \frac{1}{m-n}
\][/tex]
When we have the same term subtracted from itself, we are left with:
[tex]\[
0
\][/tex]
Thus, the simplified expression is:
[tex]\[
0
\][/tex]
So, the simplified form of the given expression
[tex]\[
\frac{1}{m-n} - \frac{m+n}{m^2 - n^2}
\][/tex]
is indeed:
[tex]\[
0
\][/tex]
