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Ta có:
\(\sqrt{6}+\sqrt[3]{6}< \sqrt{6+\sqrt{6+...+\sqrt{6}}}+\sqrt[3]{6+\sqrt[3]{6+...+\sqrt[3]{6}}}< \sqrt{6+\sqrt{6+...+\sqrt{9}}}+\sqrt[3]{6+\sqrt[3]{6+...+\sqrt[3]{8}}}\)
\(\Leftrightarrow4< \sqrt{6}+\sqrt[3]{6}< \sqrt{6+\sqrt{6+...+\sqrt{6}}}+\sqrt[3]{6+\sqrt[3]{6+...+\sqrt[3]{6}}}< 3+2\)
\(\Leftrightarrow4< \sqrt{6+\sqrt{6+...+\sqrt{6}}}+\sqrt[3]{6+\sqrt[3]{6+...+\sqrt[3]{6}}}< 5\)
1: \(=\sqrt{6}+\sqrt{6}+1=2\sqrt{6}+1\)
2: \(=\dfrac{6\left(1-\sqrt{3}\right)}{1-\sqrt{3}}+\dfrac{3\left(\sqrt{3}+1\right)}{\sqrt{3}+1}=6+3=9\)
3: \(=\sqrt{3}+1-\sqrt{3}=1\)
a) Ta có: \(\sqrt{3-2\sqrt{2}}-\sqrt{11+6\sqrt{2}}\)
\(=\sqrt{2}-1-3-\sqrt{2}\)
=-4
b) Ta có: \(\sqrt{4-2\sqrt{3}}-\sqrt{7-4\sqrt{3}}+\sqrt{19+8\sqrt{3}}\)
\(=\sqrt{3}-1-2+\sqrt{3}+4+\sqrt{3}\)
\(=3\sqrt{3}+1\)
c) Ta có: \(\sqrt{6-2\sqrt{5}}+\sqrt{9+4\sqrt{5}}-\sqrt{14-6\sqrt{5}}\)
\(=\sqrt{5}-1+\sqrt{5}-2-3+\sqrt{5}\)
\(=3\sqrt{5}-6\)
d) Ta có: \(\sqrt{11-4\sqrt{7}}+\sqrt{23-8\sqrt{7}}+\sqrt{\left(-2\right)^6}\)
\(=\sqrt{7}-2+4-\sqrt{7}+8\)
=10
1.\(\sqrt{1+2\sqrt{5}+5}=\sqrt{\left(1+\sqrt{5}\right)^2}=1+\sqrt{5}\)
2.\(\sqrt{10-4\sqrt{6}}=\sqrt{4-4\sqrt{6}+6}=\sqrt{\left(2-\sqrt{6}\right)^2}=\left|2-\sqrt{6}\right|=\sqrt{6}-2\) \(\sqrt{15-6\sqrt{6}}=\sqrt{9+6\sqrt{6}+6}=\sqrt{\left(3+\sqrt{6}\right)^2}=3+\sqrt{6}\)
=>\(\sqrt{15-6\sqrt{6}}-\sqrt{10-4\sqrt{6}}\)=\(3+\sqrt{6}-\sqrt{6}+2\)=5
3. Tương tự bằng :\(8-3\sqrt{6}\)
1) \(\sqrt{6+2\sqrt{5}}\) = \(\sqrt{1+2.1.\sqrt{5}+\sqrt{5}^2}\) = \(\sqrt{\left(1+\sqrt{5}\right)^2}\)
2) \(\sqrt{15-6\sqrt{6}}-\sqrt{10-4\sqrt{6}}\)
= \(\sqrt{3^2-2.3.\sqrt{6}+\sqrt{6}^2}\) - \(\sqrt{2^2.2.2.\sqrt{6}+\sqrt{6}^2}\)
= \(\sqrt{\left(3+\sqrt{6}\right)^2}\) - \(\sqrt{\left(2+\sqrt{6}\right)^2}\)
= \(\left|3+\sqrt{6}\right|\) - \(\left|2+\sqrt{6}\right|\)
= 3 + \(\sqrt{6}\) - 2 + \(\sqrt{6}\)
= 1 + 2\(\sqrt{6}\)
3) \(\sqrt{31-10\sqrt{6}}-\sqrt{\left(3-2\sqrt{6}\right)^2}\)
= \(\sqrt{5^2-2.5.\sqrt{6}+\sqrt{6}^2}\) - \(\sqrt{\left(3-2\sqrt{6}\right)^2}\)
= \(\sqrt{\left(5-\sqrt{6}\right)^2}\) - \(\sqrt{\left(3-2\sqrt{6}\right)^2}\)
= \(\left|5-\sqrt{6}\right|\) - \(\left|3-2\sqrt{6}\right|\)
= 5 - \(\sqrt{6}-3-2\sqrt{6}\)
= 2 - 3\(\sqrt{6}\)
Chúc bạn học tốt
6+...+\(\sqrt{6}\) cái ... là cái gì vậy quá trừu tượng