Question

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Answer 1

\(\dfrac{\sqrt{y}}{x-\sqrt{xy}}+\dfrac{\sqrt{x}}{y-\sqrt{xy}}\)

\(=\dfrac{\sqrt{y}}{\sqrt{x}\left(\sqrt{x}-\sqrt{y}\right)}+\dfrac{\sqrt{x}}{\sqrt{y}\left(\sqrt{y}-\sqrt{x}\right)}\)

\(=\dfrac{\sqrt{y}}{\sqrt{x}\left(\sqrt{x}-\sqrt{y}\right)}-\dfrac{\sqrt{x}}{\sqrt{y}\left(\sqrt{x}-\sqrt{y}\right)}\)

\(=\dfrac{y}{\sqrt{xy}\left(\sqrt{x}-\sqrt{y}\right)}-\dfrac{x}{\sqrt{xy}\left(\sqrt{x}-\sqrt{y}\right)}\)

\(=\dfrac{y-x}{\sqrt{xy}\left(\sqrt{x}-\sqrt{y}\right)}\)

\(=\dfrac{\left(\sqrt{y}-\sqrt{x}\right)\left(\sqrt{y}+\sqrt{x}\right)}{\sqrt{xy}\left(\sqrt{x}-\sqrt{y}\right)}\)

\(=\dfrac{-\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)}{\sqrt{xy}\left(\sqrt{x}-\sqrt{y}\right)}\)

\(=\dfrac{-\sqrt{x}-\sqrt{y}}{\sqrt{xy}}\)