To simplify the expression [tex]\(\frac{1}{a+b} - \frac{1}{a-b}\)[/tex], we need to follow several algebraic steps:
1. Find a common denominator for the two fractions. The denominators are [tex]\(a + b\)[/tex] and [tex]\(a - b\)[/tex]. The common denominator will be the product of these two denominators:
[tex]\[
(a+b)(a-b)
\][/tex]
2. Rewrite each fraction with the common denominator:
- For [tex]\(\frac{1}{a+b}\)[/tex], multiply the numerator and the denominator by [tex]\(a - b\)[/tex]:
[tex]\[
\frac{1}{a+b} \cdot \frac{a-b}{a-b} = \frac{a - b}{(a+b)(a-b)}
\][/tex]
- For [tex]\(\frac{1}{a-b}\)[/tex], multiply the numerator and the denominator by [tex]\(a + b\)[/tex]:
[tex]\[
\frac{1}{a-b} \cdot \frac{a+b}{a+b} = \frac{a + b}{(a+b)(a-b)}
\][/tex]
3. Combine the two fractions over the common denominator:
[tex]\[
\frac{1}{a+b} - \frac{1}{a-b} = \frac{a - b}{(a+b)(a-b)} - \frac{a + b}{(a+b)(a-b)} = \frac{(a - b) - (a + b)}{(a+b)(a-b)}
\][/tex]
4. Simplify the numerator:
- Distribute the negative sign in the second term:
[tex]\[
(a - b) - (a + b) = a - b - a - b = -2b
\][/tex]
5. Write the simplified expression:
[tex]\[
\frac{-2b}{(a+b)(a-b)}
\][/tex]
Therefore, the simplified form of the expression [tex]\(\frac{1}{a+b} - \frac{1}{a-b}\)[/tex] is:
[tex]\[
\frac{-2b}{(a+b)(a-b)}
\][/tex]
