[tex]\displaystyle\bf\\\frac{a-b}{ab^{\frac{1}{2} } -a^{\frac{1}{2} } b} =\frac{(a^{\frac{1}{2} })^{2} -(b^{\frac{1}{2} } )^{2} }{(a^{^{\frac{1}{2} } } )^{2} b^{\frac{1}{2} } -a^{\frac{1}{2} }(b^{\frac{1}{2} } )^{2} } =\frac{(a^{\frac{1}{2} } -b^{\frac{1}{2} })(a^{\frac{1}{2} }+b^{\frac{1}{2} }) }{a^{\frac{1}{2} } b^{\frac{1}{2} } \cdot(a^{\frac{1}{2} } -b^{\frac{1}{2} } )} =\\\\\\=\frac{a^{\frac{1}{2} }+b^{\frac{1}{2} } }{a^{\frac{1}{2} } b^{\frac{1}{2} } } =\frac{\sqrt{a} +\sqrt{b} }{\sqrt{ab} }[/tex]
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[tex]\displaystyle\bf\\\frac{a-b}{ab^{\frac{1}{2} } -a^{\frac{1}{2} } b} =\frac{(a^{\frac{1}{2} })^{2} -(b^{\frac{1}{2} } )^{2} }{(a^{^{\frac{1}{2} } } )^{2} b^{\frac{1}{2} } -a^{\frac{1}{2} }(b^{\frac{1}{2} } )^{2} } =\frac{(a^{\frac{1}{2} } -b^{\frac{1}{2} })(a^{\frac{1}{2} }+b^{\frac{1}{2} }) }{a^{\frac{1}{2} } b^{\frac{1}{2} } \cdot(a^{\frac{1}{2} } -b^{\frac{1}{2} } )} =\\\\\\=\frac{a^{\frac{1}{2} }+b^{\frac{1}{2} } }{a^{\frac{1}{2} } b^{\frac{1}{2} } } =\frac{\sqrt{a} +\sqrt{b} }{\sqrt{ab} }[/tex]