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

a) Cho \(a > 0.\) Chứng minh \(a + \frac{1}{a} \ge 2;\)

b) Cho \(a \ge 0,\;b \ge 0.\) Chứng minh \(\sqrt {\frac{{a + b}}{2}}  \ge \frac{{\sqrt a  + \sqrt b }}{2};\)

c) Cho \(a,b > 0.\) Chứng minh \(\sqrt a  + \sqrt b  \le \frac{a}{{\sqrt b }} + \frac{b}{{\sqrt a }};\)

d) Chứng minh \(\frac{{{x^2} + 2}}{{\sqrt {{x^2} + 1} }} \ge 2\) với mọi \(x.\)

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

Chứng minh đẳng thức

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

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) \[\left( {2\sqrt 8  - 3\sqrt 3  + 1} \right):\sqrt 6 .\]

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) \(\frac{{\sqrt 2  + \sqrt 3  + \sqrt 6  + \sqrt 8  + \sqrt {16} }}{{\sqrt 2  + \sqrt 3  + \sqrt 4 }}\)

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

Giải phương trình

a) \(\sqrt {\frac{{4x - 1}}{{x + 1}}}  = 3\);                                                                    b) \(\frac{{\sqrt {4x - 1} }}{{\sqrt {x + 1} }} = 3\);

c) \(\sqrt {49x - 98}  - 14\sqrt {\frac{{x - 2}}{{49}}}  = 3\sqrt {x - 2}  + 8\);                               d) \(\sqrt {25x - 25}  - \frac{{15}}{2}\sqrt {\frac{{x - 1}}{9}}  = 6 + \sqrt {x - 1} \).