The sum of `(a)/(a^(2)-b^(2)) and (b)/(a^(2)-b^(2))` is

Options

A. `(1)/(a-b)`
B. `(a)/(a-b)`
C. `(b)/(a-b)`
D. `(a+b)/(a-b)`

Answer

To solve the problem of finding the sum of \(\frac{a}{a^2 - b^2}\) and \(\frac{b}{a^2 - b^2}\), we can follow these steps:

### Step 1: Write down the expression
We start with the expression:
\[ \frac{a}{a^2 - b^2} + \frac{b}{a^2 - b^2} \]

### Step 2: Combine the fractions
Since both fractions have the same denominator, we can combine them:
\[ \frac{a + b}{a^2 - b^2} \]

### Step 3: Factor the denominator
Next, we recognize that \(a^2 - b^2\) can be factored using the difference of squares:
\[ a^2 - b^2 = (a + b)(a - b) \]
So we can rewrite the expression as:
\[ \frac{a + b}{(a + b)(a - b)} \]

### Step 4: Simplify the expression
Now, we can simplify the fraction by canceling the common term \(a + b\) in the numerator and the denominator (assuming \(a + b \neq 0\)):
\[ \frac{1}{a - b} \]

### Final Answer
Thus, the sum of \(\frac{a}{a^2 - b^2}\) and \(\frac{b}{a^2 - b^2}\) is:
\[ \frac{1}{a - b} \]