The expression `(x + y)^(-1).(x^(-1) + y^(-1))` is equivalent to :

Options

A. 1
B. `(xy)^(-1)`
C. `x^y`
D. `xy^(-1) + x^(-1) y`

Answer

To solve the expression \((x + y)^{-1} \cdot (x^{-1} + y^{-1})\), we will follow these steps:

### Step 1: Rewrite the expression using the property of negative exponents
We know that \(a^{-1} = \frac{1}{a}\). Therefore, we can rewrite the expression as:
\[ (x + y)^{-1} \cdot (x^{-1} + y^{-1}) = \frac{1}{x + y} \cdot \left(\frac{1}{x} + \frac{1}{y}\right) \]

### Step 2: Simplify the second part of the expression
Now, we simplify \(\frac{1}{x} + \frac{1}{y}\):
\[ \frac{1}{x} + \frac{1}{y} = \frac{y + x}{xy} = \frac{x + y}{xy} \]

### Step 3: Substitute back into the expression
Now, we substitute this back into our expression:
\[ \frac{1}{x + y} \cdot \frac{x + y}{xy} \]

### Step 4: Cancel out the common terms
Notice that \(x + y\) in the numerator and denominator cancels out:
\[ \frac{1}{xy} \]

### Step 5: Rewrite the final expression
Thus, we can write the final expression as:
\[ \frac{1}{xy} = (xy)^{-1} \]

### Conclusion
The expression \((x + y)^{-1} \cdot (x^{-1} + y^{-1})\) is equivalent to \((xy)^{-1}\).

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