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 reciprocal of \((a+b)^{-1} \cdot (a^{-1} + b^{-1})\), we will simplify the expression step by step.

### Step-by-Step Solution:

1. **Start with the expression**:
\[ (a+b)^{-1} \cdot (a^{-1} + b^{-1}) \]

2. **Rewrite \(a^{-1} + b^{-1}\)**:
Recall that \(a^{-1} = \frac{1}{a}\) and \(b^{-1} = \frac{1}{b}\). Therefore:
\[ a^{-1} + b^{-1} = \frac{1}{a} + \frac{1}{b} \]

3. **Combine the fractions**:
To combine \(\frac{1}{a} + \frac{1}{b}\), we need a common denominator, which is \(ab\):
\[ \frac{1}{a} + \frac{1}{b} = \frac{b}{ab} + \frac{a}{ab} = \frac{a+b}{ab} \]

4. **Substitute back into the expression**:
Now substitute this back into the original expression:
\[ (a+b)^{-1} \cdot \left(\frac{a+b}{ab}\right) \]

5. **Simplify the expression**:
The \((a+b)\) in the numerator and the \((a+b)^{-1}\) will cancel out:
\[ (a+b)^{-1} \cdot \frac{a+b}{ab} = \frac{1}{ab} \]

6. **Find the reciprocal**:
The reciprocal of \(\frac{1}{ab}\) is simply \(ab\):
\[ \text{Reciprocal} = ab \]

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