Rút gọn: \[F = \sqrt {3 + \sqrt 5 } + \sqrt {7 - 3\sqrt 5 } - \sqrt 2 \].

Lời giải

Ta có \[F = \sqrt {3 + \sqrt 5 } + \sqrt {7 - 3\sqrt 5 } - \sqrt 2 \]

\[ \Rightarrow \sqrt 2 .F = \sqrt 2 \,.\,\left( {\sqrt {3 + \sqrt 5 } + \sqrt {7 - 3\sqrt 5 } - \sqrt 2 } \right)\]

\[ = \sqrt {6 + 2\sqrt 5 } + \sqrt {14 - 6\sqrt 5 } - 2\]

\[ = \sqrt {{{\sqrt 5 }^2} + 2\sqrt 5 + 1} + \sqrt {{3^2} - 2\,.\,3\sqrt 5 + {{\sqrt 5 }^2}} - 2\]

\[ = \sqrt {{{\left( {\sqrt 5 + 1} \right)}^2}} + \sqrt {{{\left( {3 - \sqrt 5 } \right)}^2}} - 2\]

\[ = \left| {\sqrt 5 + 1} \right| + \left| {3 - \sqrt 5 } \right| - 2\]

\[ = \sqrt 5 + 1 + 3 - \sqrt 5 - 2\]

= 2

Vậy \(F = \frac{2}{{\sqrt 2 }} = \sqrt 2 \).