Cho \(a>0, b>0\) và biểu thức \(T = 2{\left( {a + b} \right)^{ - 1}}.{\left( {ab} \right)^{\frac{1}{2}}}.

* Hướng dẫn giải

Do \(a>0, b>0\) ta có:

\(T = 2{\left( {a + b} \right)^{ - 1}}.{\left( {ab} \right)^{\frac{1}{2}}}.{\left[ {1 + \frac{1}{4}{{\left( {\sqrt {\frac{a}{b}}  - \sqrt {\frac{b}{a}} } \right)}^2}} \right]^{\frac{1}{2}}} = \frac{{2\sqrt {ab} }}{{a + b}}.\sqrt {1 + \frac{1}{4}\left( {\frac{a}{b} - 2 + \frac{b}{a}} \right)}  = \frac{{2\sqrt {ab} }}{{a + b}}.\sqrt {1 + \frac{1}{4}.\frac{{{{\left( {a - b} \right)}^2}}}{{ab}}} \)

\( = \frac{1}{{a + b}}\sqrt {4ab + {a^2} - 2ab + {b^2}}  = \frac{{\sqrt {{{\left( {a + b} \right)}^2}} }}{{a + b}} = 1\)