Answer: [tex]\(-\frac{1}{ab}\)[/tex]
Step-by-step explanation:
1. Start with the original expression [tex]\(\frac{1}{b^2 - ab} - \frac{1}{ab - a^2}\)[/tex].
2. Notice that both denominators have common terms involving \(a\) and [tex]\(b\)[/tex]. Factor out [tex]\(-a\)[/tex] from the second denominator to make the terms more comparable:
[tex]\[ \frac{1}{b^2 - ab} - \frac{1}{ab - a^2} = \frac{1}{b^2 - ab} - \frac{1}{a(b - a)} \][/tex]
3. Next, factor [tex]\(b\)[/tex] from the first denominator:
[tex]\[ \frac{1}{b(b - a)} - \frac{1}{a(b - a)} \][/tex]
4. Now, both fractions have a common denominator, \(a(b - a)\). Combine the fractions by subtracting the numerators:
[tex]\[ \frac{a - b}{ab(b - a)} \][/tex]
5. Simplify the numerator [tex]\(a - b\) to \(-b + a\) or \(-1(b - a)\)[/tex], which allows us to cancel out the \((b - a)\) term in the numerator and denominator:
[tex]\[ \frac{-1(b - a)}{ab(b - a)} = \frac{-1}{ab} \][/tex]
6. We are left with the simplified expression:
[tex]\[ -\frac{1}{ab} \][/tex]
