(a+b-2ab/a+b)*a^2-b^2/a^2+b^2
Let's simplify the given expression step-by-step.
The expression is:
\[ \frac{(a + b - \frac{2ab}{a + b}) \cdot a^2 - b^2}{a^2 + b^2} \]
First, rewrite the first part \((a + b - \frac{2ab}{a + b})\):
- Find a common denominator for the terms inside the parentheses:
\[ \frac{(a + b)(a + b) - 2ab}{a + b} = \frac{a^2 + 2ab + b^2 - 2ab}{a + b} = \frac{a^2 + b^2}{a + b} \]
- Now substitute this back into the expression:
\[ \frac{\left(\frac{a^2 + b^2}{a + b}\right) a^2 - b^2}{a^2 + b^2} \]
- Distributing \(a^2\) within the numerator gives:
\[ \frac{\frac{a^2(a^2 + b^2)}{a + b} - b^2}{a^2 + b^2} \]
- Combine the terms in the numerator:
To combine \(\frac{a^2(a^2 + b^2)}{a + b} - b^2\),
\[ \frac{a^2(a^2 + b^2) - b^2(a + b)}{a + b} \]
- Thus, the whole expression becomes:
\[ \frac{\frac{a^2(a^2 + b^2) - b^2(a + b)}{a + b}}{a^2 + b^2} \]
- Finally, multiplying by the reciprocal:
\[ \frac{a^2(a^2 + b^2) - b^2(a + b)}{(a^2 + b^2) (a + b)} \]
Now this is a complex form, but for specific values of \(a\) and \(b\), you can substitute and calculate further.
To summarize, the simplified expression is:
\[ \frac{a^2(a^2 + b^2) - b^2(a + b)}{(a^2 + b^2)(a + b)} \]
This is as simplified as we could get without specific numerical values assigned for \(a\) and \(b\).
