Question

\[ (a+b)^{-1}(a^{-1}+b^{-1}) = a^x b^y \]

Solution

\[ = \frac{1}{a+b} \left(\frac{1}{a} + \frac{1}{b}\right) \] \[ = \frac{1}{a+b} \cdot \frac{a+b}{ab} \] \[ = \frac{1}{ab} \] \[ = a^{-1} b^{-1} \] \[ x = -1,\quad y = -1 \] \[ x + y + 2 = -1 -1 + 2 = 0 \]

Answer

\[ \boxed{0} \]

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