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

Answer

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

### Step 1: Rewrite the negative exponents
We start by rewriting the negative exponents in the expression:
\[ (a + b)^{-1} = \frac{1}{a + b} \]
\[ a^{-1} = \frac{1}{a}, \quad b^{-1} = \frac{1}{b} \]
Thus, we can rewrite the expression as:
\[ \frac{1}{a + b} \left( \frac{1}{a} + \frac{1}{b} \right) \]

### Step 2: Combine the terms inside the parentheses
Next, we combine the fractions inside the parentheses:
\[ \frac{1}{a} + \frac{1}{b} = \frac{b + a}{ab} \]
So now our expression becomes:
\[ \frac{1}{a + b} \cdot \frac{a + b}{ab} \]

### Step 3: Simplify the expression
Now, we can simplify the expression by multiplying the two fractions:
\[ \frac{1 \cdot (a + b)}{(a + b) \cdot ab} \]
Here, the \((a + b)\) in the numerator and denominator cancels out:
\[ = \frac{1}{ab} \]

### Final Answer
Thus, the simplified form of the expression \((a + b)^{-1} (a^{-1} + b^{-1})\) is:
\[ \frac{1}{ab} \]

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