Đề bài
Tính :
a] \[\sqrt 2 \left[ {\sqrt {4 - \sqrt 7 } - \sqrt {4 + \sqrt 7 } + \sqrt 2 } \right]\];
b] \[\left[ {4 - \sqrt 7 } \right]\left[ {\sqrt 2 + \sqrt {14} } \right]\sqrt {4 + \sqrt 7 } \];
c] \[\sqrt {4 + \sqrt {15} } - \sqrt {4 - \sqrt {15} } - \sqrt {2 - \sqrt 3 } \];
d] \[\sqrt {13 + 30\sqrt {2 + \sqrt {9 + 4\sqrt 2 } } } \].
Phương pháp giải - Xem chi tiết
+] Sử dụng công thức: \[\sqrt {{A^2}B} = \left| A \right|\sqrt B = \left\{ \begin{array}{l}A\;\;khi\;\;A \ge 0\\ - A\;\;\;khi\;\;A < 0\end{array} \right..\]
Lời giải chi tiết
\[\begin{array}{l}a]\;\sqrt 2 \left[ {\sqrt {4 - \sqrt 7 } - \sqrt {4 + \sqrt 7 } + \sqrt 2 } \right]\\ = \sqrt {8 - 2\sqrt 7 } - \sqrt {8 + 2\sqrt 7 } + 2\\ = \sqrt {{{\left[ {\sqrt 7 } \right]}^2} - 2\sqrt 7 + 1} - \sqrt {{{\left[ {\sqrt 7 } \right]}^2} + 2\sqrt 7 + 1} + 2\\ = \sqrt {{{\left[ {\sqrt 7 - 1} \right]}^2}} - \sqrt {{{\left[ {\sqrt 7 + 1} \right]}^2}} + 2\\ = \left| {\sqrt 7 - 1} \right| - \left| {\sqrt 7 + 1} \right| + 2\\ = \sqrt 7 - 1 - \sqrt 7 - 1 + 2 = 0\end{array}\]
\[\begin{array}{l}b]\;\left[ {4 - \sqrt 7 } \right]\left[ {\sqrt 2 + \sqrt {14} } \right]\sqrt {4 + \sqrt 7 } \\ = \dfrac{1}{2}.2\left[ {4 - \sqrt 7 } \right].\sqrt 2 \left[ {1 + \sqrt 7 } \right]\sqrt {4 + \sqrt 7 } \\ = \dfrac{1}{2}\left[ {8 - 2\sqrt 7 } \right]\left[ {1 + \sqrt 7 } \right]\sqrt {8 + 2\sqrt 7 } \\ = \dfrac{1}{2}\left[ {{{\left[ {\sqrt 7 } \right]}^2} - 2\sqrt 7 + 1} \right]\left[ {1 + \sqrt 7 } \right]\sqrt {{{\left[ {\sqrt 7 } \right]}^2} + 2\sqrt 7 + 1} \\ = \dfrac{1}{2}{\left[ {\sqrt 7 - 1} \right]^2}\left[ {1 + \sqrt 7 } \right]\left| {\sqrt 7 + 1} \right|\\ = \dfrac{1}{2}{\left[ {\sqrt 7 - 1} \right]^2}\left[ {1 + \sqrt 7 } \right]\left[ {\sqrt 7 + 1} \right]\\ = \dfrac{1}{2}{\left[ {\sqrt 7 - 1} \right]^2}{\left[ {\sqrt 7 + 1} \right]^2}\\ = \dfrac{1}{2}{\left[ {\left[ {\sqrt 7 - 1} \right]\left[ {\sqrt 7 + 1} \right]} \right]^2}\\ = \dfrac{1}{2}{\left[ {7 - 1} \right]^2} = \dfrac{1}{2}.36 = 18.\end{array}\]
\[\begin{array}{l}d]\;\sqrt {13 + 30\sqrt {2 + \sqrt {9 + 4\sqrt 2 } } } \\ = \sqrt {13 + 30\sqrt {2 + \sqrt {{{\left[ {2\sqrt 2 } \right]}^2} + 2.2\sqrt 2 + 1} } } \\ = \sqrt {13 + 30\sqrt {2 + \sqrt {{{\left[ {2\sqrt 2 + 1} \right]}^2}} } } \\ = \sqrt {13 + 30\sqrt {2 + \left| {2\sqrt 2 + 1} \right|} } \\ = \sqrt {13 + 30\sqrt {2 + 2\sqrt 2 + 1} } \\ = \sqrt {13 + 30\sqrt {{{\left[ {\sqrt 2 + 1} \right]}^2}} } \\ = \sqrt {13 + 30\left| {\sqrt 2 + 1} \right|} \\ = \sqrt {13 + 30\left[ {\sqrt 2 + 1} \right]} \\ = \sqrt {13 + 30\sqrt 2 + 30} \\ = \sqrt {{{\left[ {3\sqrt 2 } \right]}^2} + 2.3\sqrt 2 .5 + {5^2}} \\ = \sqrt {{{\left[ {3\sqrt 2 + 5} \right]}^2}} = \left| {3\sqrt 2 + 5} \right|\\ = 3\sqrt 2 + 5.\end{array}\]