Simplify the following:
x−1y−1x−1+y−1\dfrac{x^{-1}y^{-1}}{x^{-1} + y^{-1}}x−1+y−1x−1y−1
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Given,
⇒x−1y−1x−1+y−1=(xy)−1×1x−1+y−1=1xy×11x+1y=1xy×1x+yxy=1xy×xyx+y=1x+y.\Rightarrow \dfrac{x^{-1}y^{-1}}{x^{-1} + y^{-1}} = (xy)^{-1} \times \dfrac{1}{x^{-1} + y^{-1}} \\[1em] = \dfrac{1}{xy} \times \dfrac{1}{\dfrac{1}{x} + \dfrac{1}{y}} = \dfrac{1}{xy} \times \dfrac{1}{\dfrac{x + y}{xy}} \\[1em] = \dfrac{1}{xy} \times \dfrac{xy}{x + y} = \dfrac{1}{x + y}.⇒x−1+y−1x−1y−1=(xy)−1×x−1+y−11=xy1×x1+y11=xy1×xyx+y1=xy1×x+yxy=x+y1.
Hence, x−1y−1x−1+y−1=1x+y\dfrac{x^{-1}y^{-1}}{x^{-1} + y^{-1}} = \dfrac{1}{x + y}x−1+y−1x−1y−1=x+y1
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(2a-3b2)3
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