perform indicated operations.
(5ab)/(a^2-b^2)+(a+b)/(a-b)
5ab/(a+b)(a-b) + (a+b)/(a-b)
= 5ab/(a+b)(a-b) + (a+b)^2/(a+b)(a-b)
= (5ab + (a+b)^2)/(a+b)(a-b)
= (a^2 + 7ab + b^2)/(a^2 - b^2)
To perform the indicated operations, we need to simplify the given expression:
Step 1: Factor the denominators.
a^2 - b^2 can be factored as (a + b)(a - b).
a - b is already factored.
Step 2: Find the least common denominator (LCD) for the fractions.
The LCD for the fractions (5ab) / (a^2 - b^2) and (a + b) / (a - b) is (a + b)(a - b).
Step 3: Rewrite the fractions using the LCD.
(5ab)/(a^2 - b^2) can be rewritten as (5ab) / ((a + b)(a - b)).
(a + b)/(a - b) does not require any changes since its denominator is already (a - b).
Step 4: Add the fractions.
(5ab) / ((a + b)(a - b)) + (a + b) / (a - b)
To combine the fractions, we need a common denominator, which is (a + b)(a - b).
Step 5: Multiply the first fraction's numerator and denominator by (a - b).
[(5ab) * (a - b)] / [(a + b)(a - b)]
Step 6: Multiply the second fraction's numerator and denominator by (a + b).
[(a + b) * (a + b)] / [(a + b)(a - b)]
Step 7: Simplify and combine the numerators.
(5ab(a - b) + (a + b)(a + b)) / [(a + b)(a - b)]
Step 8: Expand and simplify the numerators.
(5a^2b - 5ab^2 + a^2 + 2ab + b^2) / [(a + b)(a - b)]