化简 (a^{-1}+b^{-1})(a+b)^{-1}

解答

化简

(a^{- 1} + b^{- 1}) (a + b)^{- 1}

\frac{1}{ab}

求解步骤

(a^{- 1} + b^{- 1}) (a + b)^{- 1}

a^{- 1} = \frac{1}{a}

b^{- 1} = \frac{1}{b}

= (\frac{1}{a} + \frac{1}{b}) (a + b)^{- 1}

使用指数法则:

a^{- 1} = \frac{1}{a}

(a + b)^{- 1} = \frac{1}{a + b}

= (\frac{1}{a} + \frac{1}{b}) \frac{1}{a + b}

使用分式法则:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{( \frac{1}{a} + \frac{1}{b} ) \cdot 1}{a + b}

化简

(\frac{1}{a} + \frac{1}{b}) \cdot 1 : \frac{b + a}{ab}

= \frac{\frac{b + a}{ab}}{a + b}

使用分式法则:

\frac{\frac{a}{b}}{c} = \frac{a}{b \cdot c}

= \frac{b + a}{ab ( a + b )}

使用交换律:

b + a = a + b

= \frac{a + b}{ab ( a + b )}

约分:

a + b

= \frac{1}{ab}

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