Tìm phần nguyên của: \(\sqrt{6+\sqrt{6+\sqrt{6...+\sqrt{6}}}}+\sqrt[3]{6+\sqrt[3]{6+\sqrt[3]{6+...\sqrt[3]{6}}}}\)
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NV
Nguyễn Việt Lâm
Giáo viên
16 tháng 7 2020
\(\sqrt{6+\sqrt{6+...+\sqrt{6}}}>\sqrt{6}=\sqrt{\frac{150}{25}}>\sqrt{\frac{144}{25}}=\frac{12}{5}\)
\(\sqrt[3]{6+\sqrt[3]{6+...+\sqrt[3]{6}}}>\sqrt[3]{6}=\sqrt[3]{\frac{750}{125}}>\sqrt[3]{\frac{729}{125}}=\frac{9}{5}\)
\(\Rightarrow A>\frac{12}{5}+\frac{9}{5}=\frac{21}{5}>4\)
\(\sqrt{6+\sqrt{6+...+\sqrt{6}}}< \sqrt{6+\sqrt{6+...+\sqrt{9}}}=3\)
\(\sqrt[3]{6+\sqrt[3]{6+...+\sqrt[3]{6}}}< \sqrt[3]{6+\sqrt[3]{6+...+\sqrt[3]{8}}}=2\)
\(\Rightarrow A< 3+2=5\)
\(\Rightarrow4< A< 5\Rightarrow\left[A\right]=4\)
H9
HT.Phong (9A5)
CTVHS
10 tháng 7 2023
\(\dfrac{6-\sqrt{6}}{\sqrt{6}-1}+\dfrac{6-\sqrt{6}}{\sqrt{6}}\)
\(=\dfrac{\sqrt{6}\cdot\sqrt{6}-\sqrt{6}}{\sqrt{6}-1}+\dfrac{\sqrt{6}\cdot\sqrt{6}-\sqrt{6}}{\sqrt{6}}\)
\(=\dfrac{\sqrt{6}\left(\sqrt{6}-1\right)}{\sqrt{6}-1}+\dfrac{\sqrt{6}\left(\sqrt{6}-1\right)}{\sqrt{6}}\)
\(=\dfrac{\sqrt{6}}{1}+\dfrac{\sqrt{6}-1}{1}\)
\(=\sqrt{6}+\sqrt{6}-1\)
\(=2\sqrt{6}-1\)
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\(\dfrac{1}{\sqrt{2}-\sqrt{3}}-\dfrac{3}{\sqrt{18}+2\sqrt{3}}\)
\(=\dfrac{1}{\sqrt{2}-\sqrt{3}}-\dfrac{3}{\sqrt{6}\cdot\sqrt{3}+\sqrt{6}\cdot\sqrt{2}}\)
\(=\dfrac{1}{\sqrt{2}-\sqrt{3}}-\dfrac{3}{\sqrt{6}\left(\sqrt{3}+\sqrt{2}\right)}\)
\(=\dfrac{\sqrt{6}\left(\sqrt{2}+\sqrt{3}\right)}{\sqrt{6}\left(\sqrt{2}-\sqrt{3}\right)\left(\sqrt{2}+\sqrt{3}\right)}-\dfrac{3\left(\sqrt{2}-\sqrt{3}\right)}{\sqrt{6}\left(\sqrt{2}-\sqrt{3}\right)\left(\sqrt{2}+\sqrt{3}\right)}\)
\(=\dfrac{\sqrt{6}\left(\sqrt{2}+\sqrt{3}\right)-3\left(\sqrt{2}-\sqrt{3}\right)}{-\sqrt{6}}\)
\(=\dfrac{2\sqrt{3}+3\sqrt{2}-3\sqrt{2}+3\sqrt{3}}{-\sqrt{6}}\)
\(=\dfrac{5\sqrt{3}}{-\sqrt{6}}=-\dfrac{5}{\sqrt{2}}\)