2.006,2.007 Подробно

[latex]\dfrac{( \sqrt{a} + \sqrt{b} )^2-4b}{(a-b):( \sqrt{ \frac{1}{b} }+3 \sqrt{ \frac{1}{a} } ) } : \dfrac{a+9b+6 \sqrt{ab} }{ \frac{1}{ \sqrt{b} }+\frac{1}{ \sqrt{a} } } = \\\ = \dfrac{a+b+2 \sqrt{ab}-4b}{(a-b):( \frac{1}{ \sqrt{b} }+\frac{3}{ \sqrt{a} } ) } : \dfrac{( \sqrt{a}+3 \sqrt{b} )^2 }{ \frac{1}{ \sqrt{b} }+\frac{1}{ \sqrt{a} } } = \\\ = \dfrac{a+2 \sqrt{ab}-3b}{(a-b): \frac{ \sqrt{a }+3 \sqrt{b} }{ \sqrt{ab} } } : \dfrac{( \sqrt{a}+3 \sqrt{b} )^2 }{ \frac{ \sqrt{a}+ \sqrt{b} }{ \sqrt{ab} }} } =[/latex] [latex]= \dfrac{a- \sqrt{ab}+3 \sqrt{ab}-3b}{(a-b): \frac{ \sqrt{a }+3 \sqrt{b} }{ \sqrt{ab} } } : \dfrac{\sqrt{ab}( \sqrt{a}+3 \sqrt{b} )^2 }{ \sqrt{a}+ \sqrt{b}} } = \\\ = \dfrac{ \sqrt{a} ( \sqrt{a} - \sqrt{b})+3 \sqrt{b}( \sqrt{a} - \sqrt{b}) }{(a-b)\cdot \frac{ \sqrt{ab} }{ \sqrt{a }+3 \sqrt{b} } } \cdot \dfrac{ \sqrt{a}+ \sqrt{b}}{\sqrt{ab}( \sqrt{a}+3 \sqrt{b} )^2 } } =[/latex] [latex]= \dfrac{ ( \sqrt{a} - \sqrt{b})( \sqrt{a} +3 \sqrt{b} ) }{(a-b)\cdot \frac{ \sqrt{ab} }{ \sqrt{a }+3 \sqrt{b} } } \cdot \dfrac{ \sqrt{a}+ \sqrt{b}}{\sqrt{ab}( \sqrt{a}+3 \sqrt{b} )^2 } } = \\\ =\dfrac{ ( \sqrt{a} - \sqrt{b})( \sqrt{a} +3\sqrt{b}) (\sqrt{a }+3 \sqrt{b})}{(a-b)\sqrt{ab} } \cdot \dfrac{ \sqrt{a}+ \sqrt{b}}{\sqrt{ab}( \sqrt{a}+3 \sqrt{b} )^2 } } =[/latex] [latex]=\dfrac{ ( \sqrt{a} - \sqrt{b}) (\sqrt{a }+3 \sqrt{b})^2(\sqrt{a}+ \sqrt{b})}{(a-b)\sqrt{ab} \sqrt{ab}( \sqrt{a}+3 \sqrt{b} )^2 }= \\\ =\dfrac{ ( \sqrt{a} - \sqrt{b}) (\sqrt{a}+ \sqrt{b})}{(a-b)\sqrt{ab} \sqrt{ab} }=\dfrac{ a-b}{(a-b)ab }= \dfrac{1}{ab} [/latex] [latex] \dfrac{ (\sqrt[4]{m}+ \sqrt[4]{n})^2+(\sqrt[4]{m}-\sqrt[4]{n})^2 }{2(m-n)} : \dfrac{1}{ \sqrt{m^3}- \sqrt{n^3} } -3 \sqrt{mn} = \\\ = \dfrac{ \sqrt{m}+2\sqrt[4]{mn}+ \sqrt{n} +\sqrt{m}-2\sqrt[4]{mn}+ \sqrt{n} }{2(m-n)} \cdot ( \sqrt{m^3}- \sqrt{n^3}) - \\\ -3 \sqrt{mn} =\dfrac{ 2 \sqrt{m}+ 2\sqrt{n}}{2(m-n)} \cdot ( \sqrt{m^3}- \sqrt{n^3})-3 \sqrt{mn} = \\\ =\dfrac{ \sqrt{m}+ \sqrt{n}}{(\sqrt{m}- \sqrt{n})(\sqrt{m}+ \sqrt{n})} \cdot ( \sqrt{m^3}- \sqrt{n^3})-3 \sqrt{mn} =[/latex] [latex]=\dfrac{ 1}{\sqrt{m}- \sqrt{n}} \cdot ( \sqrt{m}- \sqrt{n})(m+ \sqrt{mn}+n) -3 \sqrt{mn} = \\\ =m+ \sqrt{mn}+n -3 \sqrt{mn} =m-2 \sqrt{mn}+n =( \sqrt{m} - \sqrt{n} )^2[/latex]