Question

adding and subtracting rational expressions
what is the common denominator of \\(\frac{1}{a} + \frac{1}{b}\\) in the complex fraction \\(\frac{\frac{1}{a} - \frac{1}{b}}{\frac{1}{a} + \frac{1}{b}}\\)?
options:
\\(a - b\\)
\\(a + b\\)
\\(ab\\)
\\(a^2b^2\\)

Explanation:

Step1: Recall common denominator rule

To find the common denominator of two fractions \(\frac{1}{a}\) and \(\frac{1}{b}\), we use the formula for the least common denominator (LCD) of two fractions with denominators \(a\) and \(b\). The LCD of \(\frac{1}{a}\) and \(\frac{1}{b}\) is the product of the two denominators when they are coprime (or in general, the least common multiple, which for distinct variables \(a\) and \(b\) is \(ab\)).

Step2: Verify with addition

When we add \(\frac{1}{a}+\frac{1}{b}\), we rewrite it as \(\frac{b}{ab}+\frac{a}{ab}=\frac{a + b}{ab}\), so the common denominator is \(ab\). This common denominator is also used in the complex fraction \(\frac{\frac{1}{a}-\frac{1}{b}}{\frac{1}{a}+\frac{1}{b}}\) because the numerator and denominator of the complex fraction are \(\frac{1}{a}-\frac{1}{b}\) and \(\frac{1}{a}+\frac{1}{b}\), both of which use \(ab\) as their common denominator.

Answer:

\(ab\)