To solve for the common denominator of [tex]\(\frac{1}{a} + \frac{1}{b}\)[/tex] in the complex fraction [tex]\(\frac{\frac{1}{a} - \frac{1}{b}}{\frac{1}{a} + \frac{1}{b}}\)[/tex], let's carefully go through the steps:
1. Identify terms:
- The given fraction is [tex]\(\frac{\frac{1}{a} - \frac{1}{b}}{\frac{1}{a} + \frac{1}{b}}\)[/tex].
2. Find the common denominator for the sum:
- Consider the terms in the denominator of the complex fraction: [tex]\(\frac{1}{a}\)[/tex] and [tex]\(\frac{1}{b}\)[/tex].
3. Calculate the least common denominator (LCD):
- To combine [tex]\(\frac{1}{a}\)[/tex] and [tex]\(\frac{1}{b}\)[/tex], we need to find a common denominator.
4. Determine the LCD:
- The least common denominator for the fractions [tex]\(\frac{1}{a}\)[/tex] and [tex]\(\frac{1}{b}\)[/tex] is the product of [tex]\(a\)[/tex] and [tex]\(b\)[/tex].
5. Express the fractions with the LCD:
- Rewrite [tex]\(\frac{1}{a} + \frac{1}{b}\)[/tex] with the common denominator [tex]\(ab\)[/tex]:
[tex]\[
\frac{1}{a} = \frac{b}{ab}, \quad \frac{1}{b} = \frac{a}{ab}
\][/tex]
- Therefore,
[tex]\[
\frac{1}{a} + \frac{1}{b} = \frac{b}{ab} + \frac{a}{ab} = \frac{a + b}{ab}
\][/tex]
6. Conclusion:
- The common denominator of [tex]\(\frac{1}{a} + \frac{1}{b}\)[/tex] is [tex]\(ab\)[/tex].
Therefore, the common denominator of [tex]\(\frac{1}{a} + \frac{1}{b}\)[/tex] in the given complex fraction is:
[tex]\[ ab \][/tex]
