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\(\sqrt{\frac{9}{\sqrt{14+4\sqrt{6}}}}-\sqrt{\frac{9}{\sqrt{14-4\sqrt{6}}}}\)
\(=\sqrt{\frac{9}{\sqrt{\left(\sqrt{12}\right)^2+2\cdot\sqrt{12}\cdot\sqrt{2}+\left(\sqrt{2}\right)^2}}}-\sqrt{\frac{9}{\sqrt{\left(\sqrt{12}\right)^2-2\cdot\sqrt{12}\cdot\sqrt{2}+\left(\sqrt{2}\right)^2}}}\)
\(=\sqrt{\frac{9}{\sqrt{12}+\sqrt{2}}}-\sqrt{\frac{9}{\sqrt{12}-\sqrt{2}}}\)
\(=\frac{3}{\sqrt{\sqrt{12}+\sqrt{2}}}-\frac{3}{\sqrt{\sqrt{12}-\sqrt{2}}}=\frac{3\left(\sqrt{\sqrt{12}-\sqrt{2}}\right)-3\left(\sqrt{\sqrt{12}+\sqrt{2}}\right)}{\left(\sqrt{\sqrt{12}+\sqrt{2}}\right)\left(\sqrt{\sqrt{12}-\sqrt{2}}\right)}\)
\(=\frac{3\sqrt{\sqrt{12}-\sqrt{2}}-3\sqrt{\sqrt{12}+\sqrt{2}}}{\left(\sqrt{\sqrt{12}+\sqrt{2}}\right)\left(\sqrt{\sqrt{12}-\sqrt{2}}\right)}\)
\(=\frac{3\sqrt{\sqrt{12}-\sqrt{2}}-3\sqrt{\sqrt{12}+\sqrt{2}}}{\sqrt{12-2}}\)
\(=\frac{3\sqrt{\sqrt{12}-\sqrt{2}}-3\sqrt{\sqrt{12}+\sqrt{2}}}{\sqrt{10}}\)
\(=\frac{3\left(\sqrt{2\sqrt{3}-\sqrt{2}}-\sqrt{2\sqrt{3}+\sqrt{2}}\right)}{\sqrt{10}}\)
\(=\frac{3}{\sqrt{10}}\)
\(\sqrt{\frac{9}{\sqrt{14+4\sqrt{6}}}}-\sqrt{\frac{9}{\sqrt{14-4\sqrt{6}}}}\)
\(=\sqrt{\frac{9}{\sqrt{\left(\sqrt{12}\right)^2+2\cdot\sqrt{12}\cdot\sqrt{2}+\left(\sqrt{2}\right)^2}}}-\sqrt{\frac{9}{\sqrt{\left(\sqrt{12}\right)^2-2\cdot\sqrt{12}\cdot\sqrt{2}+\left(\sqrt{2}\right)^2}}}\)
\(=\sqrt{\frac{9}{\sqrt{12}+\sqrt{2}}}-\sqrt{\frac{9}{\sqrt{12}-\sqrt{2}}}\)
\(=\frac{3}{\sqrt{\sqrt{12}+\sqrt{2}}}-\frac{3}{\sqrt{\sqrt{12}-\sqrt{2}}}=\frac{3\left(\sqrt{\sqrt{12}-\sqrt{2}}\right)-3\left(\sqrt{\sqrt{12}+\sqrt{2}}\right)}{\left(\sqrt{\sqrt{12}+\sqrt{2}}\right)\left(\sqrt{\sqrt{12}-\sqrt{2}}\right)}\)
\(=\frac{3\sqrt{\sqrt{12}-\sqrt{2}}-3\sqrt{\sqrt{12}+\sqrt{2}}}{\left(\sqrt{\sqrt{12}+\sqrt{2}}\right)\left(\sqrt{\sqrt{12}-\sqrt{2}}\right)}\)
\(=\frac{3\sqrt{\sqrt{12}-\sqrt{2}}-3\sqrt{\sqrt{12}+\sqrt{2}}}{\sqrt{12-2}}\)\(=\frac{3\sqrt{\sqrt{12}-\sqrt{2}}-3\sqrt{\sqrt{12}+\sqrt{2}}}{\sqrt{10}}\)
\(=\frac{3\left(\sqrt{2\sqrt{3}-\sqrt{2}}-\sqrt{2\sqrt{3}+\sqrt{2}}\right)}{\sqrt{10}}\)
bí....!!!
\(A=4-\sqrt{21-8\sqrt{5}}=4-\sqrt{4^2-8\sqrt{5}+\left(\sqrt{5}\right)^2}.\)
\(A=4-\sqrt{\left(4-\sqrt{5}\right)^2}=4-\left(4-\sqrt{5}\right)\)
=> \(A=\sqrt{5}\)
Ý anh là so sánh đúng ko ạ?
15) Bình phương hai vế,ta cần so sánh: \(\left(\frac{5}{4}\sqrt{2}\right)^2\text{ và }\left(\frac{2}{3}\sqrt{7}\right)^2\Leftrightarrow\frac{25}{8}\text{ và }\frac{28}{9}\)
Dễ thấy \(\frac{25}{8}>\frac{28}{9}\Rightarrow\frac{5}{4}\sqrt{2}>\frac{2}{3}\sqrt{7}\)
16) \(\sqrt{15}-\sqrt{14}=\frac{1}{\sqrt{15}+\sqrt{14}}< \frac{1}{\sqrt{14}+\sqrt{13}}=\sqrt{14}-\sqrt{13}\)
Xíu em làm tiếp,tắm đã
17/ Tương tự câu 16,18
18) \(\sqrt{9}-\sqrt{7}=\frac{2}{\sqrt{9}+\sqrt{7}};\sqrt{7}-\sqrt{5}=\frac{2}{\sqrt{7}+\sqrt{5}}\)
Dễ thấy \(\sqrt{9}+\sqrt{7}>\sqrt{7}+\sqrt{5}\Rightarrow\sqrt{9}-\sqrt{7}< \sqrt{7}-\sqrt{5}\)
13)Ta có: \(2\sqrt{6}=\sqrt{4.6}=\sqrt{24}>\sqrt{23}\Rightarrow-2\sqrt{6}< -\sqrt{23}\)
14)\(\sqrt{111}-7< \sqrt{121}-7=11-7=4\)
:v Thứ tự ngộ nhỉ?
1. Đặt A =\(\sqrt{\frac{129}{16}+\sqrt{2}}\)
\(\sqrt{16}\)A = \(\sqrt{129+16\sqrt{2}}\)
4A = \(\sqrt{\left(8\sqrt{2}+1\right)^2}\)
4A = \(8\sqrt{2}+1\)
⇒ A = \(\frac{\text{}8\sqrt{2}+1}{4}\)= \(2\sqrt{2}\) + \(\frac{1}{4}\)
2. Đặt B = \(\sqrt{\frac{289+4\sqrt{72}}{16}}\)
\(\sqrt{16}\)B = \(\sqrt{289+24\sqrt{2}}\)
4B = \(\sqrt{\left(12\sqrt{2}+1\right)^2}\)
4B = \(12\sqrt{2}+1\)
⇒ B = \(\frac{12\sqrt{2}+1}{4}\)= \(3\sqrt{2}+\frac{1}{4}\)
3. \(\sqrt{2-\sqrt{3}}\). \(\left(\sqrt{6}+\sqrt{2}\right)\)
= \(\sqrt{2-\sqrt{3}}\). \(\sqrt{2}.\left(\sqrt{3}+1\right)\)
= \(\sqrt{4-2\sqrt{3}}\) . \(\left(\sqrt{3}+1\right)\)
= \(\sqrt{\left(\sqrt{3}-1\right)^2}\) . \(\left(\sqrt{3}+1\right)\)
= \(\left(\sqrt{3}-1\right)\). \(\left(\sqrt{3}+1\right)\)
= \(\left(\sqrt{3}\right)^2\) - 12
= 3 - 1
= 2
4. \(\left(\sqrt{21}+7\right)\). \(\sqrt{10-2\sqrt{21}}\)
= \(\left(\sqrt{21}+7\right)\) . \(\sqrt{\left(\sqrt{7}-\sqrt{3}\right)^2}\)
= \(\sqrt{7}\left(\sqrt{3}+\sqrt{7}\right)\) . \(\left(\sqrt{7}-\sqrt{3}\right)\)
= \(\sqrt{7}\) \(\left[\left(\sqrt{7}\right)^2-\left(\sqrt{3}\right)^2\right]\)
= \(\sqrt{7}\) . (7 - 3)
= 4\(\sqrt{7}\)
5. \(2.\left(\sqrt{10}-\sqrt{2}\right)\). \(\sqrt{4+\sqrt{6-2\sqrt{5}}}\)
= \(2.\left(\sqrt{10}-\sqrt{2}\right)\) . \(\sqrt{4+\sqrt{5}-1}\)
= \(2.\left(\sqrt{10}-\sqrt{2}\right)\) . \(\sqrt{3+\sqrt{5}}\)
= \(\left(\sqrt{10}-\sqrt{2}\right)\) . \(\sqrt{12+4\sqrt{5}}\)
= \(\left(\sqrt{10}-\sqrt{2}\right)\) . \(\left(\sqrt{10}+\sqrt{2}\right)\)
= \(\left(\sqrt{10}\right)^2-\left(\sqrt{2}\right)^2\)
= 10 - 2
= 8
6. \(\left(4\sqrt{2}+\sqrt{30}\right)\). \(\left(\sqrt{5}-\sqrt{3}\right)\). \(\sqrt{4-\sqrt{15}}\)
= \(\sqrt{2}\)\(\left(4+\sqrt{15}\right)\) . \(\left(\sqrt{5}-\sqrt{3}\right)\) . \(\sqrt{4-\sqrt{15}}\)
= \(\left(4+\sqrt{15}\right)\) . \(\left(\sqrt{5}-\sqrt{3}\right)\) . \(\sqrt{8-2\sqrt{15}}\)
= \(\left(4+\sqrt{15}\right)\) . \(\left(\sqrt{5}-\sqrt{3}\right)\) . \(\left(\sqrt{5}-\sqrt{3}\right)\)
= \(\left(4+\sqrt{15}\right)\) . \(\left(\sqrt{5}-\sqrt{3}\right)^2\)
= \(\left(4+\sqrt{15}\right)\). \(\left(8-2\sqrt{15}\right)\)
= 32 - \(8\sqrt{15}\) + \(8\sqrt{15}\) - 30
= 2
7. \(\left(7-\sqrt{14}\right)\) . \(\sqrt{9-2\sqrt{14}}\)
= \(\sqrt{7}\) \(\left(\sqrt{7}-\sqrt{2}\right)\). \(\left(\sqrt{7}-\sqrt{2}\right)\)
= \(\sqrt{7}\). \(\left(\sqrt{7}-\sqrt{2}\right)^2\)
= \(\sqrt{7}\) . \(\left(9-2\sqrt{14}\right)\)
= 9\(\sqrt{7}\) - 14\(\sqrt{2}\)
TICK MÌNH NHA!
b) \(B=\frac{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+4}{\sqrt{2}+\sqrt{3}+\sqrt{4}}\)
\(=\frac{\sqrt{2}+\sqrt{3}+\sqrt{6}+\sqrt{8}+2+2}{\sqrt{2}+\sqrt{3}+\sqrt{4}}\)
\(=\frac{\sqrt{2}+\sqrt{3}+\sqrt{4}+\sqrt{4}+\sqrt{6}+\sqrt{8}}{\sqrt{2}+\sqrt{3}+\sqrt{4}}\)
\(=\frac{\left(\sqrt{2}+\sqrt{3}+\sqrt{4}\right)+\left(\sqrt{4}+\sqrt{6}+\sqrt{8}\right)}{\sqrt{2}+\sqrt{3}+\sqrt{4}}\)
\(=\frac{\left(\sqrt{2}+\sqrt{3}+\sqrt{4}\right)+\sqrt{2}.\left(\sqrt{2}+\sqrt{3}+\sqrt{4}\right)}{\sqrt{2}+\sqrt{3}+\sqrt{4}}\)
\(=\frac{\left(\sqrt{2}+\sqrt{3}+\sqrt{4}\right)\left(\sqrt{2}+1\right)}{\sqrt{2}+\sqrt{3}+\sqrt{4}}=\sqrt{2}+1\)
c) \(C=\sqrt{3+2\sqrt{2}}+\sqrt{6-4\sqrt{2}}\)
\(=\sqrt{2+2\sqrt{2}+1}+\sqrt{4-4\sqrt{2}+2}\)
\(=\sqrt{\left(\sqrt{2}+1\right)^2}+\sqrt{\left(2-\sqrt{2}\right)^2}\)
\(=\left|\sqrt{2}+1\right|+\left|2-\sqrt{2}\right|\)
\(=\sqrt{2}+1+2-\sqrt{2}=3\)
\(\sqrt{3+\sqrt{5}}-\sqrt{4+\sqrt{7}}+\frac{\sqrt{14}}{2}\)
\(\approx1,581\)
1.
\(\sqrt{\frac{2+\sqrt{3}}{2}}\\ =\frac{\sqrt{2+\sqrt{3}}}{\sqrt{2}}\\ =\frac{\sqrt{4+2\sqrt{3}}}{2}\\ =\frac{\sqrt{\left(1+\sqrt{3}\right)^2}}{2}\\ =\frac{1+\sqrt{3}}{2}\)
2.
\(\sqrt{\frac{14+5\sqrt{3}}{2}}\\ =\frac{\sqrt{14+5\sqrt{3}}}{\sqrt{2}}\\ =\frac{\sqrt{28+10\sqrt{3}}}{2}\\ =\frac{\sqrt{\left(5+\sqrt{3}\right)^2}}{2}\\ =\frac{5+\sqrt{3}}{2}\)
