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[tex] \frac{ a }{ \sqrt{b} - \sqrt{a} } \div ( \frac{ \sqrt{b} }{ \sqrt{b} - \sqrt{a} } - \frac{ \sqrt{b} + \sqrt{a} }{ \sqrt{b} } ) = \\ = \frac{ a}{ \sqrt{b} - \sqrt{a} } \div \frac{( \sqrt{b} ) {}^{2} - ( \sqrt{b} + \sqrt{a}) ( \sqrt{b} - \sqrt{a}) }{ \sqrt{b} ( \sqrt{b} - \sqrt{a}) } = \\ = \frac{ a }{ \sqrt{b} - \sqrt{a} } \times \frac{ \sqrt{b}( \sqrt{b} - \sqrt{a}) }{b - b + a} = \frac{ a \sqrt{b} }{a} = \sqrt{b} [/tex]

[tex]\displaystyle\frac{a}{\sqrt{b} -\sqrt{a} } \div\bigg(\frac{\sqrt{b} }{\sqrt{b} -\sqrt{a} }- \frac{\sqrt{b}+\sqrt{a} }{\sqrt{b} } \bigg)=\\\\=\frac{a}{\sqrt{b} -\sqrt{a} } \div\frac{b-(\sqrt{b} -\sqrt{a} )(\sqrt{b}+\sqrt{a} )}{\sqrt{b}(\sqrt{b}-\sqrt{a} ) } =\\\\=\frac{a}{\sqrt{b} -\sqrt{a} } \div\frac{b-(b-a)}{\sqrt{b}(\sqrt{b}-\sqrt{a} ) } =\frac{a}{\sqrt{b} -\sqrt{a}}\div\frac{a}{\sqrt{b}(\sqrt{b}-\sqrt{a})} =\\\\=\frac{a}{\sqrt{b}-\sqrt{a} }\times\frac{\sqrt{b}(\sqrt{b}-\sqrt{a})}{a} =\sqrt{b}[/tex]