Đáp án:

\(\begin{array}{l}
e,\\
E = a - 1\\
f,\\
F = 1\\
g,\\
G =  - \dfrac{{x + y}}{{x - y}}\\
h,\\
H = \dfrac{{\sqrt {ab} }}{{\sqrt a  + \sqrt b }}\\
i,\\
I = 0
\end{array}\)

Giải thích các bước giải:

 Ta có:

\(\begin{array}{l}
e,\\
E = \dfrac{{\sqrt a  - 1}}{{a\sqrt a  - a + \sqrt a }}:\dfrac{1}{{{a^2} + \sqrt a }}\\
 = \dfrac{{\sqrt a  - 1}}{{\sqrt a .\left( {a - \sqrt a  + 1} \right)}}:\dfrac{1}{{\sqrt a .\left( {a\sqrt a  + 1} \right)}}\\
 = \dfrac{{\sqrt a  - 1}}{{\sqrt a .\left( {a - \sqrt a  + 1} \right)}}:\dfrac{1}{{\sqrt a .\left( {{{\sqrt a }^3} + {1^3}} \right)}}\\
 = \dfrac{{\sqrt a  - 1}}{{\sqrt a .\left( {a - \sqrt a  + 1} \right)}}:\dfrac{1}{{\sqrt a .\left( {\sqrt a  + 1} \right).\left( {{{\sqrt a }^2} - \sqrt a .1 + {1^2}} \right)}}\\
 = \dfrac{{\sqrt a  - 1}}{{\sqrt a .\left( {a - \sqrt a  + 1} \right)}}:\dfrac{1}{{\sqrt a \left( {\sqrt a  + 1} \right)\left( {a - \sqrt a  + 1} \right)}}\\
 = \dfrac{{\sqrt a  - 1}}{{\sqrt a .\left( {a - \sqrt a  + 1} \right)}}.\sqrt a .\left( {\sqrt a  + 1} \right)\left( {a - \sqrt a  + 1} \right)\\
 = \left( {\sqrt a  - 1} \right)\left( {\sqrt a  + 1} \right)\\
 = {\sqrt a ^2} - {1^2}\\
 = a - 1\\
f,\\
F = \left( {\dfrac{{\sqrt x  - \sqrt y }}{{x - y}} + \dfrac{{\sqrt {xy} }}{{\sqrt x  + \sqrt y }}} \right):\dfrac{{\sqrt {xy}  + 1}}{{\sqrt x  + \sqrt y }}\\
 = \left( {\dfrac{{\sqrt x  - \sqrt y }}{{{{\sqrt x }^2} - {{\sqrt y }^2}}} + \dfrac{{\sqrt {xy} }}{{\sqrt x  + \sqrt y }}} \right):\dfrac{{\sqrt {xy}  + 1}}{{\sqrt x  + \sqrt y }}\\
 = \left( {\dfrac{{\sqrt x  - \sqrt y }}{{\left( {\sqrt x  - \sqrt y } \right)\left( {\sqrt x  + \sqrt y } \right)}} + \dfrac{{\sqrt {xy} }}{{\sqrt x  + \sqrt y }}} \right):\dfrac{{\sqrt {xy}  + 1}}{{\sqrt x  + \sqrt y }}\\
 = \left( {\dfrac{1}{{\sqrt x  + \sqrt y }} + \dfrac{{\sqrt {xy} }}{{\sqrt x  + \sqrt y }}} \right):\dfrac{{\sqrt {xy}  + 1}}{{\sqrt x  + \sqrt y }}\\
 = \dfrac{{1 + \sqrt {xy} }}{{\sqrt x  + \sqrt y }}:\dfrac{{\sqrt {xy}  + 1}}{{\sqrt x  + \sqrt y }}\\
 = \dfrac{{1 + \sqrt {xy} }}{{\sqrt x  + \sqrt y }}.\dfrac{{\sqrt x  + \sqrt y }}{{\sqrt {xy}  + 1}}\\
 = 1\\
g,\\
G = \dfrac{x}{{\sqrt {xy}  + y}} + \dfrac{y}{{\sqrt {xy}  - x}} - \dfrac{{x + y}}{{\sqrt {xy} }}\\
 = \dfrac{x}{{\sqrt y .\left( {\sqrt x  + \sqrt y } \right)}} + \dfrac{y}{{\sqrt x .\left( {\sqrt y  - \sqrt x } \right)}} - \dfrac{{x + y}}{{\sqrt {xy} }}\\
 = \dfrac{x}{{\sqrt y .\left( {\sqrt x  + \sqrt y } \right)}} - \dfrac{y}{{\sqrt x .\left( {\sqrt x  - \sqrt y } \right)}} - \dfrac{{x + y}}{{\sqrt {xy} }}\\
 = \dfrac{{x.\sqrt x .\left( {\sqrt x  - \sqrt y } \right) - y.\sqrt y .\left( {\sqrt x  + \sqrt y } \right) - \left( {x + y} \right)\left( {\sqrt x  - \sqrt y } \right)\left( {\sqrt x  + \sqrt y } \right)}}{{\sqrt {xy} \left( {\sqrt x  - \sqrt y } \right)\left( {\sqrt x  + \sqrt y } \right)}}\\
 = \dfrac{{x.{{\sqrt x }^2} - x.\sqrt {xy}  - \left( {y.\sqrt {xy}  + y.{{\sqrt y }^2}} \right) - \left( {x + y} \right).\left( {{{\sqrt x }^2} - {{\sqrt y }^2}} \right)}}{{\sqrt {xy} \left( {\sqrt x  - \sqrt y } \right)\left( {\sqrt x  + \sqrt y } \right)}}\\
 = \dfrac{{{x^2} - x\sqrt {xy}  - y\sqrt {xy}  - {y^2} - \left( {x + y} \right)\left( {x - y} \right)}}{{\sqrt {xy} \left( {\sqrt x  - \sqrt y } \right)\left( {\sqrt x  + \sqrt y } \right)}}\\
 = \dfrac{{{x^2} - x\sqrt {xy}  - y\sqrt {xy}  - {y^2} - \left( {{x^2} - {y^2}} \right)}}{{\sqrt {xy} \left( {\sqrt x  - \sqrt y } \right)\left( {\sqrt x  + \sqrt y } \right)}}\\
 = \dfrac{{{x^2} - x\sqrt {xy}  - y\sqrt {xy}  - {y^2} - {x^2} + {y^2}}}{{\sqrt {xy} \left( {\sqrt x  - \sqrt y } \right)\left( {\sqrt x  + \sqrt y } \right)}}\\
 = \dfrac{{ - x\sqrt {xy}  - y\sqrt {xy} }}{{\sqrt {xy} \left( {{{\sqrt x }^2} - {{\sqrt y }^2}} \right)}}\\
 = \dfrac{{ - \sqrt {xy} .\left( {x + y} \right)}}{{\sqrt {xy} \left( {x - y} \right)}}\\
 =  - \dfrac{{x + y}}{{x - y}}\\
h,\\
H = \dfrac{{a - b}}{{\sqrt a  - \sqrt b }} - \dfrac{{\sqrt {{a^3}}  - \sqrt {{b^3}} }}{{a - b}}\\
 = \dfrac{{{{\sqrt a }^2} - {{\sqrt b }^2}}}{{\sqrt a  - \sqrt b }} - \dfrac{{{{\sqrt a }^3} - {{\sqrt b }^3}}}{{{{\sqrt a }^2} - {{\sqrt b }^2}}}\\
 = \dfrac{{\left( {\sqrt a  - \sqrt b } \right)\left( {\sqrt a  + \sqrt b } \right)}}{{\sqrt a  - \sqrt b }} - \dfrac{{\left( {\sqrt a  - \sqrt b } \right)\left( {{{\sqrt a }^2} + \sqrt a .\sqrt b  + {{\sqrt b }^2}} \right)}}{{\left( {\sqrt a  - \sqrt b } \right)\left( {\sqrt a  + \sqrt b } \right)}}\\
 = \left( {\sqrt a  + \sqrt b } \right) - \dfrac{{a + \sqrt {ab}  + b}}{{\sqrt a  + \sqrt b }}\\
 = \dfrac{{{{\left( {\sqrt a  + \sqrt b } \right)}^2} - \left( {a + \sqrt {ab}  + b} \right)}}{{\sqrt a  + \sqrt b }}\\
 = \dfrac{{{{\sqrt a }^2} + 2\sqrt {ab}  + {{\sqrt b }^2} - \left( {a + \sqrt {ab}  + b} \right)}}{{\sqrt a  + \sqrt b }}\\
 = \dfrac{{a + 2\sqrt {ab}  + b - a - \sqrt {ab}  - b}}{{\sqrt a  + \sqrt b }}\\
 = \dfrac{{\sqrt {ab} }}{{\sqrt a  + \sqrt b }}\\
i,\\
I = \dfrac{{{{\left( {\sqrt x  - \sqrt y } \right)}^2} + 4\sqrt {xy} }}{{\sqrt x  + \sqrt y }} - \dfrac{{x - y}}{{\sqrt x  - \sqrt y }}\\
 = \dfrac{{\left( {{{\sqrt x }^2} - 2\sqrt {xy}  + {{\sqrt y }^2}} \right) + 4\sqrt {xy} }}{{\sqrt x  + \sqrt y }} - \dfrac{{{{\sqrt x }^2} - {{\sqrt y }^2}}}{{\sqrt x  - \sqrt y }}\\
 = \dfrac{{{{\sqrt x }^2} + 2\sqrt {xy}  + {{\sqrt y }^2}}}{{\sqrt x  + \sqrt y }} - \dfrac{{\left( {\sqrt x  - \sqrt y } \right)\left( {\sqrt x  + \sqrt y } \right)}}{{\sqrt x  - \sqrt y }}\\
 = \dfrac{{{{\left( {\sqrt x  + \sqrt y } \right)}^2}}}{{\sqrt x  + \sqrt y }} - \left( {\sqrt x  + \sqrt y } \right)\\
 = \left( {\sqrt x  + \sqrt y } \right) - \left( {\sqrt x  + \sqrt y } \right)\\
 = 0
\end{array}\)

\(\begin{array}{l}
e,\\
E = \dfrac{{\sqrt a  - 1}}{{a\sqrt a  - a + \sqrt a }}:\dfrac{1}{{{a^2} + \sqrt a }}\\
 = \dfrac{{\sqrt a  - 1}}{{\sqrt a .\left( {a - \sqrt a  + 1} \right)}}:\dfrac{1}{{\sqrt a .\left( {a\sqrt a  + 1} \right)}}\\
 = \dfrac{{\sqrt a  - 1}}{{\sqrt a .\left( {a - \sqrt a  + 1} \right)}}:\dfrac{1}{{\sqrt a .\left( {{{\sqrt a }^3} + {1^3}} \right)}}\\
 = \dfrac{{\sqrt a  - 1}}{{\sqrt a .\left( {a - \sqrt a  + 1} \right)}}:\dfrac{1}{{\sqrt a .\left( {\sqrt a  + 1} \right).\left( {{{\sqrt a }^2} - \sqrt a .1 + {1^2}} \right)}}\\
 = \dfrac{{\sqrt a  - 1}}{{\sqrt a .\left( {a - \sqrt a  + 1} \right)}}:\dfrac{1}{{\sqrt a \left( {\sqrt a  + 1} \right)\left( {a - \sqrt a  + 1} \right)}}\\
 = \dfrac{{\sqrt a  - 1}}{{\sqrt a .\left( {a - \sqrt a  + 1} \right)}}.\sqrt a .\left( {\sqrt a  + 1} \right)\left( {a - \sqrt a  + 1} \right)\\
 = \left( {\sqrt a  - 1} \right)\left( {\sqrt a  + 1} \right)\\
 = {\sqrt a ^2} - {1^2}\\
 = a - 1\\
f,\\
F = \left( {\dfrac{{\sqrt x  - \sqrt y }}{{x - y}} + \dfrac{{\sqrt {xy} }}{{\sqrt x  + \sqrt y }}} \right):\dfrac{{\sqrt {xy}  + 1}}{{\sqrt x  + \sqrt y }}\\
 = \left( {\dfrac{{\sqrt x  - \sqrt y }}{{{{\sqrt x }^2} - {{\sqrt y }^2}}} + \dfrac{{\sqrt {xy} }}{{\sqrt x  + \sqrt y }}} \right):\dfrac{{\sqrt {xy}  + 1}}{{\sqrt x  + \sqrt y }}\\
 = \left( {\dfrac{{\sqrt x  - \sqrt y }}{{\left( {\sqrt x  - \sqrt y } \right)\left( {\sqrt x  + \sqrt y } \right)}} + \dfrac{{\sqrt {xy} }}{{\sqrt x  + \sqrt y }}} \right):\dfrac{{\sqrt {xy}  + 1}}{{\sqrt x  + \sqrt y }}\\
 = \left( {\dfrac{1}{{\sqrt x  + \sqrt y }} + \dfrac{{\sqrt {xy} }}{{\sqrt x  + \sqrt y }}} \right):\dfrac{{\sqrt {xy}  + 1}}{{\sqrt x  + \sqrt y }}\\
 = \dfrac{{1 + \sqrt {xy} }}{{\sqrt x  + \sqrt y }}:\dfrac{{\sqrt {xy}  + 1}}{{\sqrt x  + \sqrt y }}\\
 = \dfrac{{1 + \sqrt {xy} }}{{\sqrt x  + \sqrt y }}.\dfrac{{\sqrt x  + \sqrt y }}{{\sqrt {xy}  + 1}}\\
 = 1\\
g,\\
G = \dfrac{x}{{\sqrt {xy}  + y}} + \dfrac{y}{{\sqrt {xy}  - x}} - \dfrac{{x + y}}{{\sqrt {xy} }}\\
 = \dfrac{x}{{\sqrt y .\left( {\sqrt x  + \sqrt y } \right)}} + \dfrac{y}{{\sqrt x .\left( {\sqrt y  - \sqrt x } \right)}} - \dfrac{{x + y}}{{\sqrt {xy} }}\\
 = \dfrac{x}{{\sqrt y .\left( {\sqrt x  + \sqrt y } \right)}} - \dfrac{y}{{\sqrt x .\left( {\sqrt x  - \sqrt y } \right)}} - \dfrac{{x + y}}{{\sqrt {xy} }}\\
 = \dfrac{{x.\sqrt x .\left( {\sqrt x  - \sqrt y } \right) - y.\sqrt y .\left( {\sqrt x  + \sqrt y } \right) - \left( {x + y} \right)\left( {\sqrt x  - \sqrt y } \right)\left( {\sqrt x  + \sqrt y } \right)}}{{\sqrt {xy} \left( {\sqrt x  - \sqrt y } \right)\left( {\sqrt x  + \sqrt y } \right)}}\\
 = \dfrac{{x.{{\sqrt x }^2} - x.\sqrt {xy}  - \left( {y.\sqrt {xy}  + y.{{\sqrt y }^2}} \right) - \left( {x + y} \right).\left( {{{\sqrt x }^2} - {{\sqrt y }^2}} \right)}}{{\sqrt {xy} \left( {\sqrt x  - \sqrt y } \right)\left( {\sqrt x  + \sqrt y } \right)}}\\
 = \dfrac{{{x^2} - x\sqrt {xy}  - y\sqrt {xy}  - {y^2} - \left( {x + y} \right)\left( {x - y} \right)}}{{\sqrt {xy} \left( {\sqrt x  - \sqrt y } \right)\left( {\sqrt x  + \sqrt y } \right)}}\\
 = \dfrac{{{x^2} - x\sqrt {xy}  - y\sqrt {xy}  - {y^2} - \left( {{x^2} - {y^2}} \right)}}{{\sqrt {xy} \left( {\sqrt x  - \sqrt y } \right)\left( {\sqrt x  + \sqrt y } \right)}}\\
 = \dfrac{{{x^2} - x\sqrt {xy}  - y\sqrt {xy}  - {y^2} - {x^2} + {y^2}}}{{\sqrt {xy} \left( {\sqrt x  - \sqrt y } \right)\left( {\sqrt x  + \sqrt y } \right)}}\\
 = \dfrac{{ - x\sqrt {xy}  - y\sqrt {xy} }}{{\sqrt {xy} \left( {{{\sqrt x }^2} - {{\sqrt y }^2}} \right)}}\\
 = \dfrac{{ - \sqrt {xy} .\left( {x + y} \right)}}{{\sqrt {xy} \left( {x - y} \right)}}\\
 =  - \dfrac{{x + y}}{{x - y}}\\
h,\\
H = \dfrac{{a - b}}{{\sqrt a  - \sqrt b }} - \dfrac{{\sqrt {{a^3}}  - \sqrt {{b^3}} }}{{a - b}}\\
 = \dfrac{{{{\sqrt a }^2} - {{\sqrt b }^2}}}{{\sqrt a  - \sqrt b }} - \dfrac{{{{\sqrt a }^3} - {{\sqrt b }^3}}}{{{{\sqrt a }^2} - {{\sqrt b }^2}}}\\
 = \dfrac{{\left( {\sqrt a  - \sqrt b } \right)\left( {\sqrt a  + \sqrt b } \right)}}{{\sqrt a  - \sqrt b }} - \dfrac{{\left( {\sqrt a  - \sqrt b } \right)\left( {{{\sqrt a }^2} + \sqrt a .\sqrt b  + {{\sqrt b }^2}} \right)}}{{\left( {\sqrt a  - \sqrt b } \right)\left( {\sqrt a  + \sqrt b } \right)}}\\
 = \left( {\sqrt a  + \sqrt b } \right) - \dfrac{{a + \sqrt {ab}  + b}}{{\sqrt a  + \sqrt b }}\\
 = \dfrac{{{{\left( {\sqrt a  + \sqrt b } \right)}^2} - \left( {a + \sqrt {ab}  + b} \right)}}{{\sqrt a  + \sqrt b }}\\
 = \dfrac{{{{\sqrt a }^2} + 2\sqrt {ab}  + {{\sqrt b }^2} - \left( {a + \sqrt {ab}  + b} \right)}}{{\sqrt a  + \sqrt b }}\\
 = \dfrac{{a + 2\sqrt {ab}  + b - a - \sqrt {ab}  - b}}{{\sqrt a  + \sqrt b }}\\
 = \dfrac{{\sqrt {ab} }}{{\sqrt a  + \sqrt b }}\\
i,\\
I = \dfrac{{{{\left( {\sqrt x  - \sqrt y } \right)}^2} + 4\sqrt {xy} }}{{\sqrt x  + \sqrt y }} - \dfrac{{x - y}}{{\sqrt x  - \sqrt y }}\\
 = \dfrac{{\left( {{{\sqrt x }^2} - 2\sqrt {xy}  + {{\sqrt y }^2}} \right) + 4\sqrt {xy} }}{{\sqrt x  + \sqrt y }} - \dfrac{{{{\sqrt x }^2} - {{\sqrt y }^2}}}{{\sqrt x  - \sqrt y }}\\
 = \dfrac{{{{\sqrt x }^2} + 2\sqrt {xy}  + {{\sqrt y }^2}}}{{\sqrt x  + \sqrt y }} - \dfrac{{\left( {\sqrt x  - \sqrt y } \right)\left( {\sqrt x  + \sqrt y } \right)}}{{\sqrt x  - \sqrt y }}\\
 = \dfrac{{{{\left( {\sqrt x  + \sqrt y } \right)}^2}}}{{\sqrt x  + \sqrt y }} - \left( {\sqrt x  + \sqrt y } \right)\\
 = \left( {\sqrt x  + \sqrt y } \right) - \left( {\sqrt x  + \sqrt y } \right)\\
 = 0
\end{array}\)