Simplify: `(a^(-1) +b^(-1)) div (a^(-2)-b^(-2))`

Options

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

Answer

To simplify the expression \((a^{-1} + b^{-1}) \div (a^{-2} - b^{-2})\), we can follow these steps:

### Step 1: Rewrite the expression using positive exponents
We know that \(a^{-1} = \frac{1}{a}\) and \(b^{-1} = \frac{1}{b}\). Similarly, \(a^{-2} = \frac{1}{a^2}\) and \(b^{-2} = \frac{1}{b^2}\). Thus, we can rewrite the expression as:

\[ \frac{\frac{1}{a} + \frac{1}{b}}{\frac{1}{a^2} - \frac{1}{b^2}} \]

### Step 2: Simplify the numerator
The numerator \(\frac{1}{a} + \frac{1}{b}\) can be combined using a common denominator:

\[ \frac{1}{a} + \frac{1}{b} = \frac{b + a}{ab} \]

### Step 3: Simplify the denominator
The denominator \(\frac{1}{a^2} - \frac{1}{b^2}\) can be factored as a difference of squares:

\[ \frac{1}{a^2} - \frac{1}{b^2} = \frac{b^2 - a^2}{a^2b^2} = \frac{(b - a)(b + a)}{a^2b^2} \]

### Step 4: Substitute back into the expression
Now substituting the simplified numerator and denominator back into the expression gives us:

\[ \frac{\frac{b + a}{ab}}{\frac{(b - a)(b + a)}{a^2b^2}} \]

### Step 5: Simplify the overall fraction
When dividing by a fraction, we multiply by its reciprocal:

\[ \frac{b + a}{ab} \cdot \frac{a^2b^2}{(b - a)(b + a)} = \frac{(b + a) \cdot a^2b^2}{ab \cdot (b - a)(b + a)} \]

The \((b + a)\) terms cancel out:

\[ = \frac{a^2b}{b - a} \]

### Final Result
Thus, the simplified expression is:

\[ \frac{ab}{b - a} \]