Trục căn ở mẫu:

a)\(\frac{9}{{\sqrt 3 }}\)                                                                     b)\(\frac{3}{{\sqrt 5  - \sqrt 2 }}\);                           

c) \(\frac{{\sqrt 2  + 1}}{{\sqrt 2  - 1}}\);                                                                    d) \(\frac{{\sqrt 5  - \sqrt 3 }}{{\sqrt 5  + \sqrt 3 }}\)

e) \(\frac{{1 - a\sqrt a }}{{1 - \sqrt a }}\);                                                                     f)\(\frac{1}{{\sqrt {18}  + \sqrt 8  - 2\sqrt 2 }}\).

g) \(\frac{{\sqrt 2 }}{{1 + \sqrt 2  - \sqrt 3 }};\)                                                                   h) \(\frac{1}{{\sqrt 3  + \sqrt 2  - \sqrt 5 }}\)

a) Với \(x \ge 0\) và \(x \ne 4\), ta có:\(A = \frac{{\sqrt x  + 1}}{{\sqrt x  - 2}} + \frac{{2\sqrt x }}{{\sqrt x  + 2}} + \frac{{2 + 5\sqrt x }}{{4 - x}}\)

\( = \frac{{\left( {\sqrt x  + 1} \right)\left( {\sqrt x  + 2} \right)}}{{\left( {\sqrt x  - 2} \right)\left( {\sqrt x  + 2} \right)}} + \frac{{2\sqrt x \left( {\sqrt x  - 2} \right)}}{{\left( {\sqrt x  - 2} \right)\left( {\sqrt x  + 2} \right)}} - \frac{{2 + 5\sqrt x }}{{\left( {\sqrt x  - 2} \right)\left( {\sqrt x  + 2} \right)}}\)

\( = \frac{{x + 3\sqrt x  + 2 + 2x - 4\sqrt x  - 2 - 5\sqrt x }}{{\left( {\sqrt x  - 2} \right)\left( {\sqrt x  + 2} \right)}}\)\( = \frac{{3x - 6\sqrt x }}{{\left( {\sqrt x  - 2} \right)\left( {\sqrt x  + 2} \right)}} = \frac{{3\sqrt x \left( {\sqrt x  - 2} \right)}}{{\left( {\sqrt x  - 2} \right)\left( {\sqrt x  + 2} \right)}} = \frac{{3\sqrt x }}{{\sqrt x  + 2}}\)

b) Khi \(A = 2\) ta được \(\frac{{3\sqrt x }}{{\sqrt x  + 2}} = 2 \Leftrightarrow 3\sqrt x  = 2\left( {\sqrt x  + 2} \right) \Leftrightarrow \sqrt x  = 4 \Leftrightarrow x = 16{\rm{ }}(tm{\rm{ }}x \ge 0,x \ne 4)\)

Vậy \(x = 16\).

a) ĐK: \(x \ge 0,x \ne 1\), ta có:

\(C = \left( {\frac{{\sqrt x  - 2}}{{x - 1}} - \frac{{\sqrt x  + 2}}{{x + 2\sqrt x  + 1}}} \right).\frac{{{{\left( {1 - x} \right)}^2}}}{2} = \left( {\frac{{\left( {\sqrt x  - 2} \right)\left( {\sqrt x  + 1} \right) - \left( {\sqrt x  + 2} \right)\left( {\sqrt x  - 1} \right)}}{{\left( {\sqrt x  - 1} \right){{\left( {\sqrt x  + 1} \right)}^2}}}} \right).\frac{{{{\left( {1 - x} \right)}^2}}}{2}\)

\( = \frac{{x - \sqrt x  - 2 - x - \sqrt x  + 2}}{{\left( {\sqrt x  - 1} \right){{\left( {\sqrt x  + 1} \right)}^2}}}.\frac{{{{\left( {1 - x} \right)}^2}}}{2} = \frac{{ - 2\sqrt x }}{{\left( {\sqrt x  - 1} \right){{\left( {\sqrt x  + 1} \right)}^2}}}.\frac{{{{\left( {1 - x} \right)}^2}}}{2}\)

\( = \frac{{ - \sqrt x {{\left( {1 - x} \right)}^2}}}{{\left( {x - 1} \right)\left( {\sqrt x  + 1} \right)}} = \frac{{\sqrt x \left( {1 - x} \right)}}{{\left( {\sqrt x  + 1} \right)}} = \sqrt x \left( {1 - \sqrt x } \right)\)

b) Ta có:\(C > 0 \Leftrightarrow \sqrt x \left( {1 - \sqrt x } \right) > 0 \Leftrightarrow 1 - \sqrt x  > 0 \Leftrightarrow 1 > \sqrt x  \Rightarrow 0 \le x < 1.\)

c) Với \(x \ge 0,x \ne 1\), ta có\(C = \sqrt x \left( {1 - \sqrt x } \right) = \sqrt x  - x =  - \left( {x - \sqrt x } \right) =  - \left( {x - 2\sqrt x .\frac{1}{2} + \frac{1}{4} - \frac{1}{4}} \right) =  - {\left( {\sqrt x  - \frac{1}{2}} \right)^2} + \frac{1}{4}\)

Vì \( - {\left( {\sqrt x  - \frac{1}{2}} \right)^2} \le 0\) với mọi \(x \ge 0\) nên \( - {\left( {\sqrt x  - \frac{1}{2}} \right)^2} + \frac{1}{4} \le \frac{1}{4}\) với mọi \(x \ge 0\).

Do đó: \(C \le \frac{1}{4}\) với mọi \(x \ge 0\)

GTLN của \(C = \frac{1}{4}\) khi \(\sqrt x  - \frac{1}{2} = 0 \Leftrightarrow \sqrt x  = \frac{1}{2} \Leftrightarrow x = \frac{1}{4}.\)

Vậy GTLN của \(C = \frac{1}{4}\) khi \(x = \frac{1}{4}\).