49 + 20 căn 6 =  25 + 2.5.(2 căn 6) +24 =  (5 + 2 căn 6)²

tương tự vs 49 - 20 căn 6 = (5 - 2 căn 6)² =) căn ( 49 - 20 căn 6 ) = 5 - 2 căn 6

7 - 4 căn 3 = 4 - 4 căn 3 + 3 = (2 - căn 3)²  =) căn ( 7 - 4 căn 3 ) = 2 - căn 3

tự giải nhé

\(\dfrac{1}{\sqrt{49+20\sqrt{6}}}-\dfrac{1}{\sqrt{49-20\sqrt{6}}}+\dfrac{1}{\sqrt{7-4\sqrt{3}}}\)

\(=\dfrac{1}{\sqrt{5^2+2\cdot2\sqrt{6}\cdot5+\left(2\sqrt{6}\right)^2}}-\dfrac{1}{\sqrt{5^2-2\cdot2\sqrt{6}\cdot5+\left(2\sqrt{6}\right)^2}}+\dfrac{1}{\sqrt{2^2-2\cdot2\cdot\sqrt{3}+\left(\sqrt{3}\right)^2}}\)

\(=\dfrac{1}{\sqrt{\left(5+2\sqrt{6}\right)^2}}-\dfrac{1}{\sqrt{\left(5-2\sqrt{6}\right)^2}}+\dfrac{1}{\sqrt{\left(2-\sqrt{3}\right)^2}}\)

\(=\dfrac{1}{5+2\sqrt{6}}-\dfrac{1}{5-2\sqrt{6}}+\dfrac{1}{2-\sqrt{3}}\)

\(=\dfrac{5-2\sqrt{6}}{\left(5+2\sqrt{6}\right)\left(5-2\sqrt{6}\right)}-\dfrac{5+2\sqrt{6}}{\left(5+2\sqrt{6}\right)\left(5-2\sqrt{6}\right)}+\dfrac{2+\sqrt{3}}{\left(2-\sqrt{3}\right)\left(2+\sqrt{3}\right)}\)

\(=\dfrac{5-2\sqrt{6}-5-2\sqrt{6}}{1}+\dfrac{2+\sqrt{3}}{1}\)

\(=-4\sqrt{6}+2+\sqrt{3}\)

\(=\dfrac{1}{5+2\sqrt{6}}-\dfrac{1}{5-2\sqrt{6}}+\dfrac{1}{2-\sqrt{3}}\)

\(=5-2\sqrt{6}-5-2\sqrt{6}+2+\sqrt{3}\)

\(=2-4\sqrt{6}+\sqrt{3}\)

\(=\dfrac{1}{\sqrt{\sqrt{25}^2+2.\sqrt{25}.\sqrt{24}+\sqrt{24}^2}}-\dfrac{1}{\sqrt{\sqrt{25}^2-2.\sqrt{25}.\sqrt{24}+\sqrt{24}^2}}+\dfrac{1}{\sqrt{\sqrt{4}^2-2\sqrt{4}\sqrt{3}+\sqrt{3}^2}}\\ =\dfrac{1}{\sqrt{\left(\sqrt{25}+\sqrt{24}\right)^2}}-\dfrac{1}{\sqrt{\left(\sqrt{25}-\sqrt{24}\right)^2}}+\dfrac{1}{\sqrt{\left(\sqrt{4}-\sqrt{3}\right)^2}}\\ =\dfrac{1}{5+\sqrt{24}}-\dfrac{1}{5-\sqrt{24}}+\dfrac{1}{2-\sqrt{3}}\)

\(=\dfrac{5-\sqrt{24}}{25-24}-\dfrac{5+\sqrt{24}}{25-24}+\dfrac{2+\sqrt{3}}{4-3}\\ =5-\sqrt{24}-5-\sqrt{24}+2+\sqrt{3}\\ =2-4\sqrt{6}+\sqrt{3}\)

Ta có \(\sqrt[4]{49+20\sqrt{6}}=\sqrt[4]{25+10\sqrt{24}+24}=\sqrt[4]{\left(5+2\sqrt{6}\right)^2}\)

                               \(=\sqrt[4]{\left(\sqrt{3}+\sqrt{2}\right)^4}=\sqrt{3}+\sqrt{2}\)

Tương tự : \(\sqrt[4]{49-20\sqrt{6}}=\sqrt{3}-\sqrt{2}\) ( Do \(\sqrt{3}>\sqrt{2}\) )

Suy ra \(\sqrt[4]{49+20\sqrt{6}}+\sqrt[4]{49-20\sqrt{6}}=2\sqrt{3}\)

a) \(\sqrt[4]{49+20\sqrt{6}}+\sqrt[4]{49-20\sqrt{6}}=\sqrt[4]{25+2\sqrt{600}+24}+\sqrt[4]{25-2\sqrt{600}+24}\\ =\sqrt[4]{\left(\sqrt{25}+\sqrt{24}\right)^2}+\sqrt[4]{\left(\sqrt{25}-\sqrt{24}\right)^2}=\sqrt{\sqrt{25}+\sqrt{24}}+\sqrt{\sqrt{25}-\sqrt{24}}\\ =\sqrt{5+2\sqrt{6}}+\sqrt{5-2\sqrt{6}}=\sqrt{3+2\sqrt{6}+2}+\sqrt{3-2\sqrt{6}+2}\\ =\sqrt{\left(\sqrt{3}+\sqrt{2}\right)^2}+\sqrt{\left(\sqrt{3}-\sqrt{2}\right)^2}=\sqrt{3}+\sqrt{2}+\sqrt{3}-\sqrt{2}\\ =2\sqrt{3}\)

\(\sqrt{3}+\sqrt{2}+\sqrt{3}-\sqrt{2}\)= \(2\sqrt{3}\)

\(49+20\sqrt{6}=25+2.5.2\sqrt{6}+24=\left(5+2\sqrt{6}\right)^2=\left(3+2.\sqrt{3}\sqrt{2}+2\right)^2=\left(\sqrt{3}+\sqrt{2}\right)^4\)

\(\Leftrightarrow\sqrt[4]{49+20\sqrt{6}}=\sqrt{3}+\sqrt{2}\)

tuiwng tự \(\Leftrightarrow\sqrt[4]{49-20\sqrt{6}}=\sqrt{3}-\sqrt{2}\)

=> Cộng lại  = > dpcm

Trả lời:

\(\sqrt[4]{49+20\sqrt{6}}+\sqrt[4]{49-20\sqrt{6}}=2\sqrt{3}\)

Ta có:\(VT=\sqrt[4]{49+20\sqrt{6}}+\sqrt[4]{49-20\sqrt{6}}\)

                \(=\sqrt[4]{25+20\sqrt{6}+24}+\sqrt[4]{25-20\sqrt{6}+24}\)

                \(=\sqrt[4]{\left(5+2\sqrt{6}\right)^2}+\sqrt[4]{\left(5-2\sqrt{6}\right)^2}\)

                \(=\sqrt{5+2\sqrt{6}}+\sqrt{5-2\sqrt{6}}\)

                \(=\sqrt{3+2\sqrt{6}+2}+\sqrt{3-2\sqrt{6}+2}\)

                \(=\sqrt{\left(\sqrt{3}+\sqrt{2}\right)^2}+\sqrt{\left(\sqrt{3}-\sqrt{2}\right)^2}\)

                \(=\sqrt{3}+\sqrt{2}+\sqrt{3}-\sqrt{2}\)

                \(=2\sqrt{3}=VP\) 

Vậy \(\sqrt[4]{49+20\sqrt{6}}+\sqrt[4]{49-20\sqrt{6}}=2\sqrt{3}\)

rộp rộp

\(\frac{\left(5+2\sqrt{6}\right)\left(49-20\sqrt{6}\right)\sqrt{5-2\sqrt{6}}}{9\sqrt{3}-11\sqrt{2}}\)

\(=\frac{\left(5+2\sqrt{6}\right)\left(5-2\sqrt{6}\right)^2\sqrt{5-2\sqrt{6}}}{9\sqrt{3}-11\sqrt{2}}\)

\(=\frac{\left(5+2\sqrt{6}\right)\left(5-2\sqrt{6}\right)\sqrt{\left(5-2\sqrt{6}\right)^2.\left(5-2\sqrt{6}\right)}}{9\sqrt{3}-11\sqrt{2}}\)

\(=\frac{\left[25-\left(2\sqrt{6}\right)^2\right]\sqrt{\left(5-2\sqrt{6}\right)^3}}{9\sqrt{3}-11\sqrt{2}}\)

\(=\frac{\sqrt{125-150\sqrt{6}+360-48\sqrt{6}}}{9\sqrt{3}-11\sqrt{2}}\)

\(=\frac{\sqrt{485-198\sqrt{6}}}{9\sqrt{3}-11\sqrt{2}}\)

\(=\frac{\sqrt{243-2.9\sqrt{3}.11\sqrt{2}+242}}{9\sqrt{3}-11\sqrt{2}}\)

\(=\frac{\sqrt{\left(9\sqrt{3}-11\sqrt{2}\right)^2}}{9\sqrt{3}-11\sqrt{2}}=1\)

Bang 1 nha ban

\(\frac{\left(5+2\sqrt{6}\right).\left(49-20\sqrt{6}\right).\sqrt{5-2\sqrt{6}}}{9\sqrt{3}-11\sqrt{2}}\)

\(=\frac{\left(5+2\sqrt{6}\right).\left(25-20\sqrt{6}+24\right).\sqrt{5-2\sqrt{6}}}{9\sqrt{3}-11\sqrt{2}}\)

\(=\frac{\left(5+2\sqrt{6}\right).\left(5-2\sqrt{6}\right)^2.\sqrt{5-2\sqrt{6}}}{9\sqrt{3}-11\sqrt{2}}\)

\(=\frac{\left(25-\left(2\sqrt{6}\right)^2\right).\sqrt{\left(5-2\sqrt{6}\right)^2.\left(5-2\sqrt{6}\right)}}{9\sqrt{3}-11\sqrt{2}}\)

\(=\frac{1.\sqrt{\left(5-2\sqrt{6}\right)^3}}{9\sqrt{3}-11\sqrt{2}}\)

\(=\frac{\sqrt{125-150\sqrt{6}+360-48\sqrt{6}}}{9\sqrt{3}-11\sqrt{2}}\)

\(=\frac{\sqrt{485-198\sqrt{6}}}{9\sqrt{3}-11\sqrt{2}}\)

\(=\frac{\sqrt{243-2.9\sqrt{3}.11\sqrt{2}+242}}{9\sqrt{3}-11\sqrt{2}}\)

\(=\frac{\sqrt{\left(9\sqrt{3}-11\sqrt{2}\right)^2}}{9\sqrt{3}-11\sqrt{2}}\)

\(=\frac{9\sqrt{3}-11\sqrt{2}}{9\sqrt{3}-11\sqrt{2}}\)

\(=1\)

Học tốt 

a. \(\sqrt{49-20\sqrt{6}}-\sqrt{106+20\sqrt{6}}=\sqrt{\left(5-2\sqrt{6}\right)^2}-\sqrt{\left(10+\sqrt{6}\right)^2}=5-2\sqrt{6}-10-\sqrt{6}=-5-3\sqrt{6}\)

b. \(\sqrt{83-20\sqrt{6}}+\sqrt{62-20\sqrt{6}}=\sqrt{\left(5\sqrt{3}-2\sqrt{2}\right)^2}+\sqrt{\left(5\sqrt{2}-2\sqrt{3}\right)^2}=5\sqrt{3}-2\sqrt{2}+5\sqrt{2}-2\sqrt{3}=3\sqrt{3}+3\sqrt{2}\)

c. \(\sqrt{302-20\sqrt{6}}+\sqrt{203-20\sqrt{6}}=\sqrt{\left(10\sqrt{3}-\sqrt{2}\right)^2}+\sqrt{\left(10\sqrt{2}-\sqrt{3}\right)^2}=10\sqrt{3}-\sqrt{2}+10\sqrt{2}-\sqrt{3}=9\sqrt{3}+9\sqrt{2}\)

d. \(\sqrt{601-20\sqrt{6}}-\sqrt{154-20\sqrt{6}}=\sqrt{\left(10\sqrt{6}-1\right)^2}-\sqrt{\left(5\sqrt{6}-2\right)^2}=10\sqrt{6}-1-5\sqrt{6}+2=1+5\sqrt{6}\)