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\(\sqrt{48}-2\sqrt{75}+\sqrt{108}-\frac{1}{7}\sqrt{147}\)
\(=4\sqrt{3}-10\sqrt{3}+6\sqrt{3}-\sqrt{3}\)
\(=\sqrt{3}\left(4-10+6-1\right)\)
\(=-\sqrt{3}\)
\(\sqrt{5\sqrt{3}+5\sqrt{48-10\sqrt{7+\sqrt{48}}}}\)
\(=\sqrt{5\sqrt{3}+5\sqrt{48-10\sqrt{\left(2-\sqrt{3}\right)^2}}}\)
\(=\sqrt{5\sqrt{3}+5\sqrt{48-20+10\sqrt{3}}}\)
\(=\sqrt{5\sqrt{3}+5\sqrt{\left(5-\sqrt{3}\right)^2}}\)
\(=\sqrt{5\sqrt{3}+25-5\sqrt{3}}\)
= 5
\(\dfrac{\sqrt{3}-\sqrt{5+\sqrt{24}}+\sqrt{\sqrt{72}+11}}{\sqrt{6+\sqrt{20}}+\sqrt{2}-\sqrt{7+\sqrt{40}}}\)
\(=\dfrac{\sqrt{3}-\sqrt{\left(\sqrt{2}+\sqrt{3}\right)^2}+\sqrt{\left(3+\sqrt{2}\right)^2}}{\sqrt{\left(\sqrt{5}+1\right)^2}+\sqrt{2}-\sqrt{\left(\sqrt{2}+\sqrt{5}\right)^2}}\)
\(=\dfrac{\sqrt{3}-\sqrt{2}-\sqrt{3}+3+\sqrt{2}}{\sqrt{5}+1+\sqrt{2}-\sqrt{2}-\sqrt{5}}\)
\(=3\)
Rút gọn:
\(\dfrac{2\sqrt{3+\sqrt{5-\sqrt{13+\sqrt{48}}}}}{\sqrt{6}+\sqrt{2}}\)
= \(\dfrac{2\sqrt{3+\sqrt{5-\sqrt{13+4\sqrt{3}}}}}{\sqrt{6}+\sqrt{2}}\)
= \(\dfrac{2\sqrt{3+\sqrt{5-\sqrt{\left(1+2\sqrt{3}\right)^2}}}}{\sqrt{6}+\sqrt{2}}\)
= \(\dfrac{2\sqrt{3+\sqrt{5-\left(1+2\sqrt{3}\right)}}}{\sqrt{6}+\sqrt{2}}\)
= \(\dfrac{\sqrt{3+\sqrt{5-\left(1+2\sqrt{3}\right)}}.\left(\sqrt{6}-\sqrt{2}\right)}{2}\)
= \(\dfrac{\sqrt{\left[3+\sqrt{5-\left(1+2\sqrt{3}\right)}\right].6}-\sqrt{\left[3+\sqrt{5-\left(1+2\sqrt{3}\right)}\right].2}}{2}\)
= \(\dfrac{\sqrt{\left(3+\sqrt{5-1-2\sqrt{3}}\right).6}-\sqrt{\left(3+\sqrt{5-1-2\sqrt{3}}\right).2}}{2}\)
= \(\dfrac{\sqrt{\left(3+\sqrt{4-2\sqrt{3}}\right).6}-\sqrt{\left(3+\sqrt{4-2\sqrt{3}}\right).2}}{2}\)
= \(\dfrac{\sqrt{\left[3+\sqrt{\left(1-\sqrt{3}\right)^2}\right].6}-\sqrt{\left[3+\sqrt{\left(1-\sqrt{3}\right)^2}\right].2}}{2}\)
= \(\dfrac{\sqrt{\left(3+\sqrt{3}-1\right).6}-\sqrt{\left(3+\sqrt{3}-1\right).2}}{2}\)
= \(\dfrac{\sqrt{\left(2+\sqrt{3}\right).6}-\sqrt{\left(2+\sqrt{3}\right).2}}{2}\)
= \(\dfrac{\sqrt{12+6\sqrt{3}}-\sqrt{4+2\sqrt{3}}}{2}\)
= \(\dfrac{\sqrt{\left(3+\sqrt{3}\right)^2}-\sqrt{\left(1+\sqrt{3}\right)^2}}{2}\)
= \(\dfrac{3+\sqrt{3}-\left(1+\sqrt{3}\right)}{2}\)
= \(\dfrac{3+\sqrt{3}-1-\sqrt{3}}{2}\)
= \(\dfrac{2}{2}\)
= \(1\)
\(A=\sqrt{\sqrt{7}-\sqrt{3}}-\sqrt{\sqrt{7}+\sqrt{3}}\)
=>\(A^2=\sqrt{7}-\sqrt{3}+\sqrt{7}+\sqrt{3}-2\sqrt{4}\)
=>A^2=2căn 7-4
=>A=2căn 7-4
=>\(M=\dfrac{2\left(\sqrt{7}-2\right)}{\sqrt{7}-2}=2\)
\(A=\dfrac{1}{2\sqrt{2}}-\dfrac{2}{\sqrt{2}+1}-\dfrac{7}{2\sqrt{2}+4}\)
\(=\dfrac{\sqrt{2}}{4}-2\left(\sqrt{2}-1\right)-\dfrac{7\left(2\sqrt{2}-4\right)}{-8}\)
\(=\dfrac{\sqrt{2}}{4}-2\sqrt{2}+2-\dfrac{7\cdot2\left(\sqrt{2}-2\right)}{-8}\)
\(=-\dfrac{7\sqrt{2}}{4}+2-\dfrac{7\left(\sqrt{2}-2\right)}{-4}\)
\(=-\dfrac{7\sqrt{2}}{4}+2-\dfrac{7\sqrt{2}-14}{-4}\)
\(=-\dfrac{7\sqrt{2}}{4}+2-\dfrac{7\sqrt{2}-14}{4}\)
\(=\dfrac{-7\sqrt{2}+7\sqrt{2}-14}{4}+2\)
\(=-\dfrac{14}{4}+2\)
\(=-\dfrac{3}{2}\)
A = \(\sqrt{48}-2\sqrt{75}+\sqrt{108}-\dfrac{1}{7}\sqrt{147}\)
= \(\sqrt{16.3}-2\sqrt{25.3}+\sqrt{36.3}-\dfrac{1}{7}\sqrt{49.3}\)
= \(4\sqrt{3}-10\sqrt{3}+6\sqrt{3}-\sqrt{3}\)
= \(-\sqrt{3}\)
\(-\sqrt{3}\)