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

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

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 \(x.\)

Rút gọn rồi tính

a) \(\sqrt {{{21,8}^2} - {{18,2}^2}} ;\)             b) \(\sqrt {{{6,8}^2} - {{3,2}^2}} ;\)                     c) \(\sqrt {{{146,5}^2} - {{109.5}^2} + 27 \cdot 256} \)

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

Chứng minh đẳng thức

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

Tìm \(x\) biết

a) \(\sqrt {9x} = 15\);       b) \(\sqrt {4{x^2}} = 8\)                                            c) \(\sqrt {4(x + 1)} = \sqrt 8 \);

d) \(\sqrt {9{{(2 - 3x)}^2}} = 6:\)   e) \(\sqrt {{x^2} - 4} - \sqrt {x - 2} = 0\).