simplify the following
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![( {a}^{ - 1} + b {}^{ - 1}) \div ( {a}^{ - 2} - {b}^{ - 2} ) \\ \\ = ( {a}^{ - 1} + {b}^{ - 1} ) \div ( \: ( {{a}^{ - 1})}^{2} - ({{b}^{ - 1})}^{2} \: ) \\ \\ = ( {a}^{ - 1} + {b}^{ - 1} ) \div [ ( {a}^{ - 1} + {b}^{ - 1} )( {a}^{ - 1} - {b}^{ - 1} )]\\ \\ = 1 \div ( {a}^{ - 1} - {b}^{ - 1} ) \\ \\ = 1 \div ( \: \frac{1}{a} - \frac{1}{b \: } \: ) \\ \\ = 1 \div \frac{b - a}{ab} \\ \\ = 1 \times \frac{ab}{b - a} \\ \\ = \frac{ab}{b - a}](https://tex.z-dn.net/?f=%28+%7Ba%7D%5E%7B+-+1%7D+%2B+b+%7B%7D%5E%7B+-+1%7D%29+%5Cdiv+%28+%7Ba%7D%5E%7B+-+2%7D+-+%7Bb%7D%5E%7B+-+2%7D+%29+%5C%5C+%5C%5C+%3D+%28+%7Ba%7D%5E%7B+-+1%7D+%2B+%7Bb%7D%5E%7B+-+1%7D+%29+%5Cdiv+%28+%5C%3A+%28+%7B%7Ba%7D%5E%7B+-+1%7D%29%7D%5E%7B2%7D+-+%28%7B%7Bb%7D%5E%7B+-+1%7D%29%7D%5E%7B2%7D+%5C%3A+%29+%5C%5C+%5C%5C+%3D+%28+%7Ba%7D%5E%7B+-+1%7D+%2B+%7Bb%7D%5E%7B+-+1%7D+%29+%5Cdiv+%5B+%28+%7Ba%7D%5E%7B+-+1%7D+%2B+%7Bb%7D%5E%7B+-+1%7D+%29%28+%7Ba%7D%5E%7B+-+1%7D+-+%7Bb%7D%5E%7B+-+1%7D+%29%5D%5C%5C+%5C%5C+%3D+1+%5Cdiv+%28+%7Ba%7D%5E%7B+-+1%7D+-+%7Bb%7D%5E%7B+-+1%7D+%29+%5C%5C+%5C%5C+%3D+1+%5Cdiv+%28+%5C%3A+%5Cfrac%7B1%7D%7Ba%7D+-+%5Cfrac%7B1%7D%7Bb+%5C%3A+%7D+%5C%3A+%29+%5C%5C+%5C%5C+%3D+1+%5Cdiv+%5Cfrac%7Bb+-+a%7D%7Bab%7D+%5C%5C+%5C%5C+%3D+1+%5Ctimes+%5Cfrac%7Bab%7D%7Bb+-+a%7D+%5C%5C+%5C%5C+%3D+%5Cfrac%7Bab%7D%7Bb+-+a%7D+ "( {a}^{ - 1} + b {}^{ - 1}) \div ( {a}^{ - 2} - {b}^{ - 2} ) \\ \\ = ( {a}^{ - 1} + {b}^{ - 1} ) \div ( \: ( {{a}^{ - 1})}^{2} - ({{b}^{ - 1})}^{2} \: ) \\ \\ = ( {a}^{ - 1} + {b}^{ - 1} ) \div [ ( {a}^{ - 1} + {b}^{ - 1} )( {a}^{ - 1} - {b}^{ - 1} )]\\ \\ = 1 \div ( {a}^{ - 1} - {b}^{ - 1} ) \\ \\ = 1 \div ( \: \frac{1}{a} - \frac{1}{b \: } \: ) \\ \\ = 1 \div \frac{b - a}{ab} \\ \\ = 1 \times \frac{ab}{b - a} \\ \\ = \frac{ab}{b - a}")
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sushant2505:
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