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\(\left(\sqrt{18}+\sqrt{0,5}-3\sqrt{\frac{1}{3}}\right)-\left(\sqrt{\frac{1}{8}-\sqrt{75}}\right)\)
\(=3\sqrt{2}+\frac{\sqrt{2}}{2}-\sqrt{3}-\frac{\sqrt{2}}{4}+5\sqrt{3}\)
\(=\frac{13\sqrt{2}}{4}+4\sqrt{3}\)
\(=\frac{13\sqrt{2}+16\sqrt{3}}{4}\)
\(\left(\sqrt{18}+\sqrt{0,5}-3\sqrt{\frac{1}{3}}\right)-\left(\sqrt{\frac{1}{8}}-\sqrt{75}\right)\)
\(=\left(\sqrt{2\cdot3^2}+\sqrt{\frac{1}{2}}-3\sqrt{\frac{3}{3^2}}\right)-\left(\sqrt{\frac{2}{4^2}}-\sqrt{3\cdot5^2}\right)\)
\(=\left(3\sqrt{2}+\sqrt{\frac{2}{2^2}}-3\cdot\frac{1}{3}\sqrt{3}\right)-\left(\frac{\sqrt{2}}{4}-5\sqrt{3}\right)\)
\(=3\sqrt{2}+\frac{\sqrt{2}}{2}-\sqrt{3}-\frac{\sqrt{2}}{4}+5\sqrt{3}\)
\(=\frac{12\sqrt{2}}{4}+\frac{2\sqrt{2}}{4}-\frac{\sqrt{2}}{4}-\sqrt{3}+5\sqrt{3}\)
\(=\frac{13\sqrt{2}}{4}+4\sqrt{3}=\frac{13\sqrt{2}+16\sqrt{3}}{4}\)
22) \(\frac{1}{\sqrt{5}+\sqrt{2}}+\frac{1}{\sqrt{5}-\sqrt{2}}\)
\(=\frac{\left(\sqrt{5}-\sqrt{2}\right)+\left(\sqrt{5}+\sqrt{2}\right)}{\left(\sqrt{5}+\sqrt{2}\right)\left(\sqrt{5}-\sqrt{2}\right)}\)
\(=\frac{2\sqrt{5}}{\sqrt{5^2}-\sqrt{2^2}}\)
\(=\frac{2\sqrt{5}}{5-2}=\frac{2\sqrt{5}}{3}\)
a) Ta có: \(2\sqrt{3}+\sqrt{48}-\sqrt{75}-\sqrt{243}\)
\(=\sqrt{3}\left(2+\sqrt{16}-\sqrt{25}-\sqrt{81}\right)\)
\(=\sqrt{3}\left(2+4-5-9\right)\)
\(=-8\sqrt{3}\)
b) Ta có: \(\left(\frac{\sqrt{7}-\sqrt{14}}{1-\sqrt{2}}+\frac{\sqrt{15}-\sqrt{5}}{1-\sqrt{3}}\right):\frac{1}{\sqrt{7}+\sqrt{5}}\)
\(=\left(\frac{\sqrt{7}\left(1-\sqrt{2}\right)}{1-\sqrt{2}}-\frac{\sqrt{5}\left(1-\sqrt{3}\right)}{1-\sqrt{3}}\right)\cdot\left(\sqrt{7}+\sqrt{5}\right)\)
\(=\left(\sqrt{7}-\sqrt{5}\right)\left(\sqrt{7}+\sqrt{5}\right)\)
\(=7-5=2\)
c) Ta có: \(\left(\sqrt{3}+1\right)\sqrt{4-2\sqrt{3}}\)
\(=\left(\sqrt{3}+1\right)\cdot\sqrt{3-2\cdot\sqrt{3}\cdot1+1}\)
\(=\left(\sqrt{3}+1\right)\cdot\sqrt{\left(\sqrt{3}-1\right)^2}\)
\(=\left(\sqrt{3}+1\right)\cdot\left|\sqrt{3}-1\right|\)
\(=\left(\sqrt{3}+1\right)\left(\sqrt{3}-1\right)\)(Vì \(\sqrt{3}>1\))
\(=3-1=2\)
d) Ta có: \(5\sqrt{2}+\sqrt{18}-\sqrt{98}-\sqrt{288}\)
\(=\sqrt{2}\cdot\left(5+\sqrt{9}-\sqrt{49}-\sqrt{144}\right)\)
\(=\sqrt{2}\cdot\left(5+3-7-12\right)\)
\(=-11\sqrt{2}\)
e) Ta có: \(\left(\frac{\sqrt{3}-\sqrt{6}}{1-\sqrt{2}}+\frac{\sqrt{15}-\sqrt{5}}{1-\sqrt{3}}\right):\frac{1}{\sqrt{3}+\sqrt{5}}\)
\(=\left(\frac{\sqrt{3}\left(1-\sqrt{2}\right)}{1-\sqrt{2}}-\frac{\sqrt{5}\left(1-\sqrt{3}\right)}{1-\sqrt{3}}\right)\cdot\left(\sqrt{3}+\sqrt{5}\right)\)
\(=\left(\sqrt{3}-\sqrt{5}\right)\left(\sqrt{3}+\sqrt{5}\right)\)
\(=3-5=-2\)
g) Ta có: \(\left(\sqrt{3}-1\right)\cdot\sqrt{4+2\sqrt{3}}\)
\(=\left(\sqrt{3}-1\right)\cdot\sqrt{3+2\cdot\sqrt{3}\cdot1+1}\)
\(=\left(\sqrt{3}-1\right)\cdot\sqrt{\left(\sqrt{3}+1\right)^2}\)
\(=\left(\sqrt{3}-1\right)\cdot\left|\sqrt{3}+1\right|\)
\(=\left(\sqrt{3}-1\right)\cdot\left(\sqrt{3}+1\right)\)(Vì \(\sqrt{3}>1>0\))
\(=3-1=2\)