Đáp án:
\[\sqrt {6} \]
Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
\sqrt {5 - \sqrt {21} } - \sqrt {5 + \sqrt {21} } \\
= \sqrt {\dfrac{1}{2}\left( {10 - 2\sqrt {21} } \right)} - \sqrt {\dfrac{1}{2}\left( {10 + 2\sqrt {21} } \right)} \\
= \sqrt {\dfrac{1}{2}.\left( {7 - 2.\sqrt 7 .\sqrt 3 + 3} \right)} - \sqrt {\dfrac{1}{2}.\left( {7 + 2.\sqrt 7 .\sqrt 3 + 3} \right)} \\
= \sqrt {\dfrac{1}{2}.{{\left( {\sqrt 7 - \sqrt 3 } \right)}^2}} - \sqrt {\dfrac{1}{2}.{{\left( {\sqrt 7 + \sqrt 3 } \right)}^2}} \\
= \sqrt {\dfrac{1}{2}} .\left( {\sqrt 7 - \sqrt 3 } \right) - \sqrt {\dfrac{1}{2}} .\left( {\sqrt 7 + \sqrt 3 } \right)\\
= \dfrac{{\sqrt 2 }}{2}.\left( {\sqrt 7 - \sqrt 3 } \right) - \dfrac{{\sqrt 2 }}{2}\left( {\sqrt 7 + \sqrt 3 } \right)\\
= \dfrac{{\sqrt 2 }}{2}.\left[ {\left( {\sqrt 7 - \sqrt 3 } \right) - \left( {\sqrt 7 + \sqrt 3 } \right)} \right]\\
= \dfrac{{\sqrt 2 }}{2}.-2\sqrt 3 \\
= -\sqrt 2 .\sqrt 3 \\
= -\sqrt {6}
\end{array}\)
