\(\left(\sqrt{21}-7\right)\sqrt{5+\sqrt{21}}\)
\(=-\left(7-\sqrt{21}\right)\sqrt{5+\sqrt{21}}\)
\(=-\left(6+1-\sqrt{21}\right)\sqrt{6-1+\sqrt{21}}\)
\(=-\sqrt{7-\sqrt{21}}\sqrt{6+\left(1-\sqrt{21}\right)}\sqrt{6-\left(1-\sqrt{21}\right)}\)
\(=-\sqrt{7-\sqrt{21}}\sqrt{36-\left(1-\sqrt{21}\right)^2}\)
\(=-\sqrt{7-\sqrt{21}}\sqrt{36-1+2\sqrt{21}-21}\)
\(=-\sqrt{7-\sqrt{21}}\sqrt{14+2\sqrt{21}}\)
\(=-\sqrt{7-\sqrt{21}}\sqrt{2\left(7+\sqrt{21}\right)}\)
\(=-\sqrt{2}\sqrt{\left(7-\sqrt{21}\right)\left(7+\sqrt{21}\right)}\)
\(=-\sqrt{2}\sqrt{49-21}\)
\(=-\sqrt{2}\sqrt{28}\)
\(=-\sqrt{56}=-2\sqrt{14}\)
Chúc bn học tốt!
Đặt \(A=\left(\sqrt{21}-7\right)\sqrt{5+\sqrt{21}}\)
\(\sqrt{2}A=\sqrt{10+2\sqrt{7.3}}\left(\sqrt{21}-7\right)\)
\(=\left(\sqrt{7}+\sqrt{3}\right)\left(\sqrt{21}-7\right)\Rightarrow A=\dfrac{\sqrt{147}-7+\sqrt{63}-7\sqrt{3}}{\sqrt{2}}\)
\(=\dfrac{\sqrt{294}-7\sqrt{2}+\sqrt{126}-7\sqrt{6}}{2}\)
Sửa đề:\(\left(\sqrt{21}-\sqrt{7}\right)\cdot\sqrt{5+\sqrt{21}}\)
Ta có: \(\left(\sqrt{21}-\sqrt{7}\right)\cdot\sqrt{5+\sqrt{21}}\)
\(=\left(\sqrt{21}-\sqrt{7}\right)\cdot\dfrac{\sqrt{10+2\sqrt{21}}}{\sqrt{2}}\)
\(=\sqrt{7}\left(\sqrt{7}-\sqrt{3}\right)\left(\sqrt{7}+\sqrt{3}\right)\cdot\dfrac{1}{\sqrt{2}}\)
\(=\dfrac{4\sqrt{7}}{\sqrt{2}}=2\sqrt{14}\)
