Find the value of reciprocal of `(a + b)^(-1) (a^(-1) + b^(-1))`

Options

A. ab
B. a/b
C. 1/ab
D. `(ab)^2`

Answer

To find the value of the expression \((a + b)^{-1} (a^{-1} + b^{-1})\), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ (a + b)^{-1} (a^{-1} + b^{-1}) \]
We can rewrite \(a^{-1} + b^{-1}\) using a common denominator:
\[ a^{-1} + b^{-1} = \frac{b + a}{ab} = \frac{a + b}{ab} \]

### Step 2: Substitute back into the expression
Now, substitute this back into the original expression:
\[ (a + b)^{-1} \left(\frac{a + b}{ab}\right) \]

### Step 3: Simplify the expression
Now we can simplify this:
\[ = (a + b)^{-1} \cdot \frac{a + b}{ab} \]
The \((a + b)\) terms cancel out:
\[ = \frac{1}{ab} \]

### Step 4: Find the reciprocal
The reciprocal of \(\frac{1}{ab}\) is simply:
\[ ab \]

### Final Answer
Thus, the value of the reciprocal of \((a + b)^{-1} (a^{-1} + b^{-1})\) is:
\[ \boxed{ab} \]