$ \left( {\sqrt a  + \dfrac{{b - \sqrt {ab} }}{{\sqrt a  + \sqrt b }}} \right):\left( {\dfrac{a}{{\sqrt {ab}  + b}} + \dfrac{b}{{\sqrt {ab}  - a}} - \dfrac{{a + b}}{{\sqrt {ab} }}} \right)\\  = \dfrac{{a + \sqrt {ab}  + b - \sqrt {ab} }}{{\sqrt a  + \sqrt b }}:\left( {\dfrac{a}{{\sqrt b \left( {\sqrt a  + \sqrt b } \right)}} + \dfrac{b}{{\sqrt a \left( {\sqrt b  - \sqrt a } \right)}} - \dfrac{{a + b}}{{\sqrt {ab} }}} \right)\\  = \dfrac{{a + b}}{{\sqrt a  + \sqrt b }}:\left( {\dfrac{{a\sqrt a \left( {\sqrt b  - \sqrt a } \right) + b\sqrt b \left( {\sqrt a  + \sqrt b } \right) - \left( {b - a} \right)\left( {b + a} \right)}}{{\sqrt {ab} \left( {\sqrt a  + \sqrt b } \right)\left( {\sqrt b  - \sqrt a } \right)}}} \right)\\  = \dfrac{{a + b}}{{\sqrt a  + \sqrt b }}:\left( {\dfrac{{a\sqrt {ab}  - {a^2} + b\sqrt {ab}  + {b^2} - \left( {{b^2} - {a^2}} \right)}}{{\sqrt {ab} \left( {b - a} \right)}}} \right)\\  = \dfrac{{a + b}}{{\sqrt a  + \sqrt b }}:\dfrac{{\sqrt {ab} \left( {a + b} \right)}}{{\sqrt {ab} \left( {b - a} \right)}} = \dfrac{{a + b}}{{\sqrt a  + \sqrt b }}.\dfrac{{b - a}}{{a + b}} = \dfrac{{b - a}}{{\sqrt a  + \sqrt b }} = \sqrt b  - \sqrt a  $  

`(\sqrt(a)+(b-\sqrt(ab))/(\sqrt(a)+\sqrt(b))):(a/(b+\sqrt(ab))+b/(\sqrt(ab)-a)-(a+b)/(\sqrt(ab)))`

`=((a\sqrt(ab)+a+b-\sqrt(ab))/(\sqrt(a)+\sqrt(b))):(a/(b+\sqrt(ab))+b/(\sqrt(ab)-a)-(a+b)/(\sqrt(ab)))`

`=(a\sqrt(ab)+a+b-\sqrt(ab))/(\sqrt(a)+\sqrt(b)):(a\sqrt(a)(\sqrt(b)-\sqrt(a))+b\sqrt(b)(\sqrt(b)+\sqrt(a))+(a+b)(a-b))/((b-a)\sqrt(ab))`

`=(a\sqrt(ab)+a+b-\sqrt(ab))/(\sqrt(a)+\sqrt(b)):(a\sqrt(ab)-a^2+b^2+b\sqrt(ab)+a^2-b^2)/((b-a)\sqrt(ab))`

`=(a+b)/(\sqrt(a)+\sqrt(b)):((a+b)\sqrt(ab))/((b-a)\sqrt(ab))`

`=(a+b)/(\sqrt(a)+\sqrt(b)):(a+b)/(b-a)`

`=(a+b)/(\sqrt(a)+\sqrt(b)).(b-a)/(a+b)`

`=(b-a)/(\sqrt(a)+\sqrt(b))`

`=((\sqrt(a)+\sqrt(b))(\sqrt(b)-\sqrt(a)))/(\sqrt(a)+\sqrt(b))`

`=\sqrt(b)-\sqrt(a)`