Simplifying the Expression: (a+b2ab/a+b)*a^2b^2/a^2+b^2
This article will guide you through simplifying the algebraic expression:
*(a+b2ab/a+b)a^2b^2/a^2+b^2
Step 1: Simplify the first part of the expression
Let's focus on the first part: (a+b2ab/a+b)

Find a common denominator: The common denominator for the terms inside the parenthesis is (a+b).
 (a+b) can be written as (a+b)/(a+b)
 2ab/(a+b) remains the same

Combine the terms: Now we have:
 (a+b)/(a+b) + (b)/(a+b)  (2ab)/(a+b)

Simplify the numerator:
 (a+b+b2ab)/(a+b)
 (a+2b2ab)/(a+b)
Step 2: Substitute the simplified expression back into the original equation
Now we have: [(a+2b2ab)/(a+b)] * a^2  b^2 / a^2 + b^2
Step 3: Factor the difference of squares
Notice that a^2  b^2 is a difference of squares. It can be factored as (a+b)(ab)
Step 4: Simplify the entire expression
Combining all the steps, the expression becomes:
[(a+2b2ab)/(a+b)] * (a+b)(ab) / (a^2+b^2)
 (a+b) cancels out from the numerator and denominator.
This leaves us with: (a+2b2ab)(ab) / (a^2+b^2)
Conclusion
The simplified expression is (a+2b2ab)(ab) / (a^2+b^2). This process involved combining terms, finding common denominators, factoring, and canceling common factors. Remember to always follow the order of operations and look for opportunities to simplify the expression.