Rút gọn các biểu thức :
a) \(\frac{{3 + 3\sqrt 5 - \sqrt 2 - \sqrt {10} }}{{6 + 2\sqrt 5 }}\);                                                                     b) \(\frac{{x\sqrt x + x\sqrt y - y\sqrt x - y\sqrt y }}{{x - y + y\sqrt x - y\sqrt y }}\).

a) Điều kiện : \(x \ge 0;x \ne 9\). Khi đó ta có

\(P = \frac{{2\sqrt x \left( {\sqrt x  - 3} \right) + \left( {\sqrt x  + 1} \right)\left( {\sqrt x  + 3} \right) - 3 + 11\sqrt x }}{{\left( {\sqrt x  + 3} \right)\left( {\sqrt x  - 3} \right)}} = \frac{{2x - 6\sqrt x  + x + 4\sqrt x  + 3 - 3 + 11\sqrt x }}{{\left( {\sqrt x  + 3} \right)\left( {\sqrt x  - 3} \right)}}\)

\( = \frac{{3x + 9\sqrt x }}{{\left( {\sqrt x  + 3} \right)\left( {\sqrt x  - 3} \right)}} = \frac{{3\sqrt x \left( {\sqrt x  + 3} \right)}}{{\left( {\sqrt x  + 3} \right)\left( {\sqrt x  - 3} \right)}} = \frac{{3\sqrt x }}{{\sqrt x  - 3}}\)

b) Ta có \(x = \frac{{7 + 4\sqrt 3 }}{4} = {\left( {\frac{{2 + \sqrt 3 }}{2}} \right)^2} \Rightarrow \sqrt x  = \frac{{2 + \sqrt 3 }}{2}\).
Do đó \({\rm{P}} = \frac{{3 \cdot \frac{{2 + \sqrt 3 }}{2}}}{{\frac{{2 + \sqrt 3 }}{2} - 3}} = \frac{{6 + 3\sqrt 3 }}{2} \cdot \frac{2}{{\sqrt 3  - 4}}\)\( = \frac{{6 + 3\sqrt 3 }}{{\sqrt 3  - 4}} = \frac{{\left( {6 + 3\sqrt 3 } \right)\left( {\sqrt 3  + 4} \right)}}{{\left( {\sqrt 3  - 4} \right)\left( {\sqrt 3  + 4} \right)}}\; = \frac{{6\sqrt 3  + 24 + 9 + 12\sqrt 3 }}{{3 - 16}} = \frac{{ - \left( {33 + 18\sqrt 3 } \right)}}{{13}}\begin{array}{*{20}{r}}{}&\;\\{}&.\end{array}\)

a) Ta có \(\sqrt {25{{(3x - 1)}^2}}  = 10\)

\(5\left| {3x - 1} \right| = 10\)

\(\left| {3x - 1} \right| = 2\)

\(3x - 1 = 2\) hoặc \(3x - 1 =  - 2\)

\(x = 1\) \(x =  - \frac{1}{3}\)

b) \(\frac{{\sqrt x  + 3}}{{\sqrt x  - 3}} = \frac{{\sqrt x  + 5}}{{\sqrt x  - 2}}\left( {\rm{*}} \right)\)
Điều kiện : \(x \ge 0;x \ne 4;x \ne 9\). Khi đó ta có

\(\begin{array}{l}\left( {\sqrt x  + 3} \right)\left( {\sqrt x  - 2} \right) = \left( {\sqrt x  + 5} \right)\left( {\sqrt x  - 3} \right)\left( {\rm{*}} \right)\\x + \sqrt x  - 6 = x + 2\sqrt x  - 15\end{array}\)

\(\begin{array}{l} - \sqrt x  =  - 9\\\sqrt x  = 9\end{array}\)

\(x = 81{\rm{\;}}\)(thỏa mãn điều kiện)