Tính

a)\[\sqrt {2\frac{7}{{81}}} \] và \[\frac{{\sqrt 6 }}{{\sqrt {150} }};\]      b) \[\left( {5\sqrt 7  + 7\sqrt 5 } \right):\sqrt {35} ;\,\] c) \[\left( {2\sqrt 8  - 3\sqrt 3  + 1} \right):\sqrt 6 .\]

Cho \(a \ge 0,b \ge 0,c \ge 0\). Chứng minh rằng:

a) \(\frac{{a + b}}{2} \ge \sqrt {ab} \) (bất đẳng thức Cauchy);

b) \(a + b + c \ge \sqrt {ab}  + \sqrt {bc}  + \sqrt {ca} \);

c) \(a + b + \frac{1}{2} \ge \sqrt a  + \sqrt b \).

Tìm \(x - y\) biết: \(\frac{1}{{\sqrt x }} + \frac{1}{{\sqrt y }} = 4 - \sqrt x - \sqrt y \)

Chứng minh đẳng thức

a) \(\sqrt {9 - \sqrt {17} }  \cdot \sqrt {9 + \sqrt {17} }  = 8\);                                                      b) \(\left( {\frac{1}{{5 - 2\sqrt 6 }} + \frac{2}{{5 + 2\sqrt 6 }}} \right)\left( {15 + 2\sqrt 6 } \right) = 201\)

Rút gọn biểu thức

a) \(\frac{{\sqrt {15}  - \sqrt 6 }}{{\sqrt {35}  - \sqrt {14} }}\);                                                    b) \(\frac{{\sqrt {10}  + \sqrt {15} }}{{\sqrt 8  + \sqrt {12} }}\)

c) \(\frac{{x + \sqrt {xy} }}{{y + \sqrt {xy} }}\) d) \(\frac{{\sqrt a  + a\sqrt b  - \sqrt b  - b\sqrt a }}{{ab - 1}}\)

e) \(\frac{{2\sqrt {15}  - 2\sqrt {10}  + \sqrt 6  - 3}}{{2\sqrt 5  - 2\sqrt {10}  - \sqrt 3  + \sqrt 6 }}\)                                                                    f) \(\frac{{\sqrt 2  + \sqrt 3  + \sqrt 6  + \sqrt 8  + \sqrt {16} }}{{\sqrt 2  + \sqrt 3  + \sqrt 4 }}\)

Tính: \(\left( {\frac{1}{2}\sqrt {\frac{1}{2}} - \frac{3}{2}\sqrt {4,5} + \frac{2}{5}\sqrt {50} } \right):\frac{4}{{15}}\sqrt {\frac{1}{8}} \).

Rút gọn biểu thức

a) \(\,\frac{{x\sqrt x  + y\sqrt y }}{{\sqrt x  + \sqrt y }} - {\left( {\sqrt x  - \sqrt y } \right)^2};\)

b) \(\,\,\sqrt {\frac{{x - 2\sqrt x  + 1}}{{x + 2\sqrt x  + 1}}} ,\,\,\left( {x \ge 0} \right)\)

c)\(\,\frac{{x - 1}}{{\sqrt y  - 1}}.\sqrt {\frac{{{{\left( {y - 2\sqrt y  + 1} \right)}^2}}}{{{{\left( {x - 1} \right)}^4}}}} ,\,\,\left( {x \ne 1,y \ne 1,y > 0} \right)\).

Thực hiện phép tính

a) \(\sqrt {\frac{1}{8}}  \cdot \sqrt 2  \cdot \sqrt {125}  \cdot \sqrt {\frac{1}{5}} ;\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\sqrt {\sqrt 2  - 1}  \cdot \sqrt {\sqrt 2  + 1} \).

b) \(\sqrt {{{(\sqrt 2  - 3)}^2}}  \cdot \sqrt {11 + 6\sqrt 2 } ;\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\sqrt {{{(\sqrt 3  - 3)}^2}}  \cdot \sqrt {\frac{1}{{3 - \sqrt 3 }}} \).

\(c)\, - \frac{2}{3}\sqrt {\frac{{{{(a - b)}^2}{b^5}}}{c}}  \cdot \frac{9}{4}\sqrt {\frac{{{c^3}}}{{2(a - b)}}} \sqrt {98b} \)

d) \(\left( {\sqrt 6  - 3\sqrt 3  + 5\sqrt 2  - \frac{1}{2}\sqrt 1 } \right)2\sqrt 6 \)

e) \(\left( {\sqrt {ab}  + 2\sqrt {\frac{b}{a}}  - \sqrt {\frac{a}{b} + \sqrt {\frac{1}{{ab}}} } } \right)\sqrt {ab} \).

g) \(\left( {\frac{{am}}{b}\sqrt {\frac{n}{m}}  - \frac{{ab}}{n}\sqrt {mn}  + \frac{{{a^2}}}{{{b^2}}}\sqrt {\frac{m}{n}} } \right){a^2}{b^2} \cdot \sqrt {\frac{n}{m}} \).