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\(\sqrt{49-5\sqrt{96}-\sqrt{49}+5\sqrt{96}}\)
\(=\sqrt{49-\sqrt{49}}\)
\(=\sqrt{49-7}\)
\(=\sqrt{42}\)
NẾU SAI BN THÔNG CẢM NHA
![](https://rs.olm.vn/images/avt/0.png?1311)
\(\sqrt{15-6\sqrt{6}}+\sqrt{35-12\sqrt{6}}=\sqrt{\left(3-\sqrt{6}\right)^2}+\sqrt{\left(3\sqrt{3}-2\sqrt{2}\right)^2}\)
\(=3-\sqrt{6}+3\sqrt{3}-2\sqrt{2}\)
\(\sqrt{17-3\sqrt{32}}+\sqrt{17+3\sqrt{32}}=\sqrt{\left(3-2\sqrt{2}\right)^2}+\sqrt{\left(3+2\sqrt{2}\right)^2}\)
\(=3-2\sqrt{2}+3+2\sqrt{2}=6\)
\(\sqrt{49-5\sqrt{96}}+\sqrt{49+5\sqrt{96}}=\sqrt{\left(5-2\sqrt{6}\right)^2}+\sqrt{\left(5+2\sqrt{6}\right)^2}\)
\(=5-2\sqrt{6}+5+2\sqrt{6}=10\)
\(\sqrt{13-\sqrt{160}}+\sqrt{53+4\sqrt{90}}=\sqrt{\left(2\sqrt{2}-\sqrt{5}\right)^2}+\sqrt{\left(3\sqrt{5}+2\sqrt{2}\right)^2}\)
\(=2\sqrt{2}-\sqrt{5}+3\sqrt{5}+2\sqrt{2}=2\sqrt{5}+4\sqrt{2}\)
a: \(\sqrt{15-6\sqrt{6}}+\sqrt{35-12\sqrt{6}}\)
\(=3-\sqrt{6}+3\sqrt{3}-2\sqrt{2}\)
b: \(\sqrt{17-3\sqrt{32}}+\sqrt{17+3\sqrt{32}}\)
\(=3-2\sqrt{2}+3+2\sqrt{2}\)
=6
c: Ta có: \(\sqrt{49-5\sqrt{96}}+\sqrt{49+5\sqrt{96}}\)
\(=5-2\sqrt{6}+5+2\sqrt{6}\)
=10
d: Ta có: \(\sqrt{13-\sqrt{160}}+\sqrt{53+4\sqrt{90}}\)
\(=\sqrt{13-4\sqrt{10}}+\sqrt{53+4\sqrt{90}}\)
\(=2\sqrt{2}-\sqrt{5}+3\sqrt{5}+2\sqrt{2}\)
\(=2\sqrt{5}+4\sqrt{2}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
câu đầu có \(3-12\sqrt{6}< 0\) nên không căn được nên đề bạn sai
\(\sqrt{31-8\sqrt{15}}+\sqrt{24-6\sqrt{15}}\)
\(=\sqrt{4^2-2.4.\sqrt{15}+\left(\sqrt{15}\right)^2}+\sqrt{\left(\sqrt{15}\right)^2-2.\sqrt{15}.3+3^2}\)
\(=\sqrt{\left(4-\sqrt{15}\right)^2}+\sqrt{\left(\sqrt{15}-3\right)^2}=\left|4-\sqrt{15}\right|+\left|\sqrt{15}-3\right|\)
\(=4-\sqrt{15}+\sqrt{15}-3=1\)
\(\sqrt{49-5\sqrt{96}}-\sqrt{49+5\sqrt{96}}=\sqrt{49-20\sqrt{6}}-\sqrt{49+20\sqrt{6}}\)
\(=\sqrt{5^2-2.5.2\sqrt{6}+\left(2\sqrt{6}\right)^2}-\sqrt{5^2+2.5.4\sqrt{6}+\left(2\sqrt{6}\right)^2}\)
\(=\sqrt{\left(5-2\sqrt{6}\right)^2}-\sqrt{\left(5+2\sqrt{6}\right)^2}=\left|5-2\sqrt{6}\right|-\left|5+2\sqrt{6}\right|\)
\(=5-2\sqrt{6}-5-2\sqrt{6}=-4\sqrt{6}\)
\(\sqrt{31-8\sqrt{15}}+\sqrt{24-6\sqrt{15}}\)
\(=4-\sqrt{15}+\sqrt{15}-3\)
=1
![](https://rs.olm.vn/images/avt/0.png?1311)
a, \(\sqrt{15-6\sqrt{6}}+\sqrt{35-12\sqrt{6}}\)
= \(\sqrt{3^2-2.3.\sqrt{6}+\left(\sqrt{6}\right)^2}+\sqrt{6^2-2.6.\sqrt{6}+\left(\sqrt{6}\right)^2}\)
= \(\sqrt{\left(3-\sqrt{6}\right)^2}+\sqrt{\left(6-\sqrt{6}\right)^2}\)
= \(\left|3-\sqrt{6}\right|+\left|6-\sqrt{6}\right|\)
= \(3-\sqrt{6}+6-\sqrt{6}\)
= \(9-2\sqrt{6}\)
b. Đặt B = \(\sqrt{17-3\sqrt{32}}+\sqrt{17+3\sqrt{32}}\)
Nhận xét : B > 0 , bình phương hai vế ta được :
\(B^2=\left(\sqrt{17-3\sqrt{32}}\right)^2+\left(\sqrt{17+3\sqrt{32}}\right)^2\)
\(B^2=17-3\sqrt{32}+17+3\sqrt{32}+2\sqrt{\left(17-3\sqrt{32}\right)\left(17+3\sqrt{32}\right)}\)
\(B^2=34+2\sqrt{17^2-\left(3\sqrt{32}\right)^2}\)
\(B^2=34+2\sqrt{289-288}\)
\(B^2=34+2=36\)
=> \(B=\pm\sqrt{36}\) mà B > 0 nên \(B=\sqrt{36}=6\)
c, Đặt C = \(\sqrt{49-5\sqrt{96}}+\sqrt{49+5\sqrt{96}}\)
Nhận xét : C > 0 , bình phương hai vế ta đươc :
\(C^2=\left(\sqrt{49-5\sqrt{96}}\right)^2+\left(\sqrt{49+5\sqrt{96}}\right)^2\)
\(C^2=49-5\sqrt{96}+49+5\sqrt{96}+2\sqrt{\left(49-5\sqrt{96}\right)\left(49+5\sqrt{96}\right)}\)
\(C^2=98+2\sqrt{49^2-\left(5\sqrt{96}\right)^2}\)
\(C^2=98+2\sqrt{2401-2400}\)
\(C^2=98+2=100\)
=> \(C=\pm\sqrt{100}\) mà C > 0 nên \(C=\sqrt{100}=10\)
a) Ta có: \(\sqrt{15-6\sqrt{6}}+\sqrt{35-12\sqrt{6}}\)
\(=\sqrt{9-2\cdot3\cdot\sqrt{6}+6}+\sqrt{27-2\cdot3\sqrt{3}\cdot2\sqrt{2}+8}\)
\(=\sqrt{\left(3-\sqrt{6}\right)^2}+\sqrt{\left(3\sqrt{3}-2\sqrt{2}\right)^2}\)
\(=\left|3-\sqrt{6}\right|+\left|3\sqrt{3}-2\sqrt{2}\right|\)
\(=3-\sqrt{6}+3\sqrt{3}-2\sqrt{2}\)(Vì \(\left\{{}\begin{matrix}3>\sqrt{6}\\3\sqrt{3}>2\sqrt{2}\end{matrix}\right.\))
b) Ta có: \(\sqrt{17-3\sqrt{32}}+\sqrt{17+3\sqrt{32}}\)
\(=\frac{\sqrt{34-6\sqrt{32}}+\sqrt{34+6\sqrt{32}}}{\sqrt{2}}\)
\(=\frac{\sqrt{18-2\cdot3\sqrt{2}\cdot4+16}+\sqrt{18+2\cdot3\sqrt{2}\cdot4+16}}{\sqrt{2}}\)
\(=\frac{\sqrt{\left(3\sqrt{2}-4\right)^2}+\sqrt{\left(3\sqrt{2}+4\right)^2}}{\sqrt{2}}\)
\(=\frac{\left|3\sqrt{2}-4\right|+\left|3\sqrt{2}+4\right|}{\sqrt{2}}\)
\(=\frac{3\sqrt{2}-4+3\sqrt{2}+4}{\sqrt{2}}\)(Vì \(3\sqrt{2}>4>0\))
\(=\frac{6\sqrt{2}}{\sqrt{2}}=6\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(A=\left(2-\sqrt{3}\right)\sqrt{4+2.2.\sqrt{3}+3}=\left(2-\sqrt{3}\right)\left(2+\sqrt{3}\right)=1\)
các câu còn lại làm tương tự nhé bạn !
![](https://rs.olm.vn/images/avt/0.png?1311)
\(\sqrt{49-5\sqrt{96}}-\sqrt{49+5\sqrt{96}}\)
\(=\sqrt{49-20\sqrt{6}}-\sqrt{49+20\sqrt{6}}\)
\(=\sqrt{\left(5-2\sqrt{6}\right)^2}-\sqrt{\left(5+2\sqrt{6}\right)^2}\)
\(=\left(5-2\sqrt{6}\right)-\left(5+2\sqrt{6}\right)\)
\(=-4\sqrt{6}\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(\sqrt{5-\sqrt{21}}=\sqrt{\frac{1}{2}}.\sqrt{10-2\sqrt{21}}=\sqrt{\frac{1}{2}}.\sqrt{3-2\sqrt{3}\sqrt{7}+7}=\sqrt{\frac{1}{2}}\sqrt{\left(\sqrt{7}-\sqrt{3}\right)^2}=\sqrt{\frac{1}{2}}.\sqrt{7}-\sqrt{\frac{1}{2}}.\sqrt{3}=\sqrt{3,5}-\sqrt{1,5}\)
\(\sqrt{7+3\sqrt{5}}=\sqrt{\frac{1}{2}\left(14+2.3\sqrt{5}\right)}=\sqrt{\frac{1}{2}\left(5+2.3\sqrt{5}+3^2\right)}=\sqrt{\frac{1}{2}\left(3+\sqrt{5}\right)^2}=\sqrt{\frac{1}{2}}\left(3+\sqrt{5}\right)=\sqrt{4,5}+\sqrt{2,5}\)
\(\sqrt{49+5\sqrt{96}}=\sqrt{49+2.2.5\sqrt{6}}=\sqrt{2^2.6+2.2.5\sqrt{6}+5^2}=\sqrt{\left(5+2\sqrt{6}\right)^2}=5+2\sqrt{6}\)
\(\sqrt{5-\sqrt{21}}=\frac{\sqrt{10-2\sqrt{21}}}{\sqrt{2}}=\frac{\sqrt{7-2\sqrt{7\cdot3}+3}}{\sqrt{2}}=\frac{\sqrt{\left(\sqrt{7}-\sqrt{3}\right)^2}}{\sqrt{2}}=\frac{\sqrt{7}-\sqrt{3}}{\sqrt{2}}\)
\(\sqrt{7+3\sqrt{5}}=\frac{\sqrt{14+6\sqrt{5}}}{\sqrt{2}}=\frac{\sqrt{9+2\cdot3\sqrt{5}+4}}{\sqrt{2}}=\frac{\sqrt{\left(3+\sqrt{5}\right)^2}}{\sqrt{2}}=\frac{3+\sqrt{5}}{\sqrt{2}}\)
\(\sqrt{49+5\sqrt{96}}=\sqrt{49+5\sqrt{4\cdot24}}=\sqrt{25+2\cdot5\sqrt{24}+24}=\sqrt{\left(5+\sqrt{24}\right)^2}=5+\sqrt{24}\)
\(\sqrt{51-7\sqrt{8}}=\sqrt{51-7\sqrt{2^2\cdot2}}=\sqrt{49-2\cdot7\sqrt{2}+2}=\sqrt{\left(7+\sqrt{2}\right)^2}=7+\sqrt{2}\)
\(\sqrt{28+5\sqrt{12}}=\sqrt{28+5\sqrt{2^2\cdot3}}=\sqrt{25+2\cdot5\sqrt{3}+3}=\sqrt{\left(5+\sqrt{3}\right)^2}=5+\sqrt{3}\)
\(\sqrt{12-3\sqrt{12}}=\sqrt{12-3\sqrt{2^2\cdot3}}=\sqrt{9-2\cdot3\sqrt{3}+3}=\sqrt{\left(3+\sqrt{3}\right)^2}=3+\sqrt{3}=\sqrt{3}\left(\sqrt{3}+1\right)\)
Chúc bạn học tốt nha.