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\(\frac{1}{2}\sqrt{48}-2\sqrt{75}-\sqrt{54}+5\sqrt{1\frac{1}{3}}\) =\(2\sqrt{3}-10\sqrt{3}-3\sqrt{6}+\frac{10\sqrt{3}}{3}\)
=\(\frac{-24\sqrt{3}-9\sqrt{6}+10\sqrt{3}}{3}=\frac{-14\sqrt{3}-9\sqrt{6}}{3}\)
Bài 1: Thực hiện phép tính
a) Ta có: \(\frac{3+\sqrt{7}}{3-\sqrt{7}}-\frac{3-\sqrt{7}}{3+\sqrt{7}}\)
\(=\frac{\left(3+\sqrt{7}\right)^2}{\left(3-\sqrt{7}\right)\left(3+\sqrt{7}\right)}-\frac{\left(3-\sqrt{7}\right)^2}{\left(3+\sqrt{7}\right)\left(3-\sqrt{7}\right)}\)
\(=\frac{9+6\sqrt{7}+7-\left(9-6\sqrt{7}+7\right)}{9-7}\)
\(=\frac{16+6\sqrt{7}-16+6\sqrt{7}}{2}\)
\(=\frac{12\sqrt{7}}{2}=6\sqrt{7}\)
b)Sửa đề: \(\left(\frac{\sqrt{2}+5}{\sqrt{2}-5}-\frac{\sqrt{2}-5}{\sqrt{2}+5}\right):\frac{\sqrt{2}}{23}\)
Ta có: \(\left(\frac{\sqrt{2}+5}{\sqrt{2}-5}-\frac{\sqrt{2}-5}{\sqrt{2}+5}\right):\frac{\sqrt{2}}{23}\)
\(=\left(\frac{\left(\sqrt{2}+5\right)^2}{\left(\sqrt{2}-5\right)\left(\sqrt{2}+5\right)}-\frac{\left(\sqrt{2}-5\right)^2}{\left(\sqrt{2}+5\right)\left(\sqrt{2}-5\right)}\right)\cdot\frac{23}{\sqrt{2}}\)
\(=\left(\frac{27+10\sqrt{2}-\left(27-10\sqrt{2}\right)}{2-25}\right)\cdot\frac{23}{\sqrt{2}}\)
\(=\frac{27+10\sqrt{2}-27+10\sqrt{2}}{-23}\cdot\frac{23}{\sqrt{2}}\)
\(=\frac{20\sqrt{2}}{-\sqrt{2}}=-20\)
c) Ta có: \(5\sqrt{\frac{1}{5}}+\frac{1}{2}\sqrt{20}+\sqrt{5}\)
\(=\sqrt{25\cdot\frac{1}{5}}+\frac{1}{2}\cdot2\sqrt{5}+\sqrt{5}\)
\(=\sqrt{5}+\sqrt{5}+\sqrt{5}\)
\(=3\sqrt{5}\)
d) Ta có: \(\sqrt{\frac{1}{2}}+\sqrt{4.5}+12.5\)
\(=\frac{1}{\sqrt{2}}+\frac{3}{\sqrt{2}}+12.5\)
\(=2\sqrt{2}+12.5\)
e) Ta có: \(\frac{1}{2}\sqrt{48}-2\sqrt{75}-\sqrt{54}+5\sqrt{1\frac{1}{3}}\)
\(=\frac{1}{2}\cdot4\sqrt{3}-2\cdot5\sqrt{3}-3\sqrt{6}+5\cdot\sqrt{\frac{4}{3}}\)
\(=2\sqrt{3}-10\sqrt{3}-3\sqrt{6}+\frac{10}{\sqrt{3}}\)
\(=-8\sqrt{3}+\frac{10}{\sqrt{3}}-3\sqrt{6}\)
\(=\frac{-24+10}{\sqrt{3}}-\frac{9\sqrt{2}}{\sqrt{3}}\)
\(=\frac{-14-9\sqrt{2}}{\sqrt{3}}\)
a) \(\frac{x\sqrt[3]{y}+\sqrt[3]{x^2y^2}}{\sqrt[3]{x^2y^2}+y\sqrt[3]{x}}\)
\(=\frac{\sqrt[3]{x^2y}\left(\sqrt[3]{x}+\sqrt[3]{y}\right)}{\sqrt[3]{xy^2}\left(\sqrt[3]{x}+\sqrt[3]{y}\right)}=\sqrt[3]{\frac{x^2y}{xy^2}}=\sqrt[3]{\frac{x}{y}}\)
b) \(\frac{\sqrt[3]{54}-2\sqrt[3]{16}}{\sqrt[3]{54}+2\sqrt[3]{16}}\)
\(=\frac{\sqrt[3]{27.2}-2\sqrt[3]{8.2}}{\sqrt[3]{27.2}+2\sqrt[3]{8.2}}\)
\(=\frac{3\sqrt[3]{2}-4\sqrt[3]{2}}{3\sqrt[3]{2}+4\sqrt[3]{2}}=\frac{-\sqrt[3]{2}}{7\sqrt[3]{2}}=-\frac{1}{7}\)
1, \(=\frac{4\sqrt{2}\left(2-\sqrt{2}\right)}{2^2-\sqrt{2}^2}-\frac{4\sqrt{2}\left(2+\sqrt{2}\right)}{2^2-\sqrt{2}^2}\)
=\(\frac{4\sqrt{2}\left(2-\sqrt{2}\right)}{2}-\frac{4\sqrt{2}\left(2+\sqrt{2}\right)}{2}\)
=\(2\sqrt{2}\left(2-\sqrt{2}\right)-2\sqrt{2}\left(2+\sqrt{2}\right)\)
=\(4\sqrt{2}-4-4\sqrt{2}-4\)
=-8
2, =\(\sqrt{2}+\sqrt{2}-2.3\sqrt{2}+\left|1-\sqrt{2}\right|\)
= \(-4\sqrt{2}+1-\sqrt{2}\) = \(1-5\sqrt{2}\)
3, =\(9\sqrt{\frac{2.2}{3.2}}+5\sqrt{9.6}-\sqrt{\frac{1}{6}}\)
=\(3\sqrt{6}+15\sqrt{6}-\frac{1}{6}\sqrt{6}\)
=\(\frac{107}{6}\sqrt{6}\)
4, =\(\sqrt{\left(4+2\sqrt{2}\right)\left(4-2\sqrt{2}\right)}.\left(2\sqrt{2}-\sqrt{2}\right)\)
= \(\sqrt{4^2-\left(2\sqrt{2}\right)^2}.\sqrt{2}\)
= \(\sqrt{16-8}.\sqrt{2}\)
= \(\sqrt{8}.\sqrt{2}=\sqrt{16}=4\)
5, = \(\sqrt{9-2.3.\sqrt{5}+5}+\sqrt{1-2.1.\sqrt{2}+2}+\sqrt{5-2.\sqrt{2}.\sqrt{5}+2}\)
\(=\sqrt{\left(3-\sqrt{5}\right)^2}+\sqrt{(1-\sqrt{2})^2}+\sqrt{\left(\sqrt{5}-\sqrt{2}\right)^2}\)\(=\left|3-\sqrt{5}\right|+\left|1-\sqrt{2}\right|+\left|\sqrt{5}-\sqrt{2}\right|\)
\(=3-\sqrt{5}+1-\sqrt{2}+\sqrt{5}-\sqrt{2}\)
\(=4-2\sqrt{2}\)
Ta có: \(\frac{1}{3}\left(\sqrt{6}+\sqrt{5}\right)^2-\frac{1}{4}\sqrt{120}-2\sqrt{\frac{15}{2}}\)
\(=\frac{1}{3}\left(11+2\sqrt{30}\right)-\frac{\sqrt{30}}{2}-\sqrt{30}\)
\(=\frac{11}{3}+\frac{2}{3}\sqrt{30}-\frac{\sqrt{30}}{2}-\sqrt{30}\)
\(=\frac{11}{3}-\frac{5}{6}\sqrt{30}\)
\(=\frac{22-5\sqrt{30}}{6}\)
Ta có: \(\left(\frac{1}{2}\sqrt{\frac{2}{3}}-\frac{3}{4}\sqrt{54}+\frac{1}{3}\sqrt{\frac{8}{3}}\right)\div\sqrt{\frac{81}{6}}\)
\(=\left(\frac{\sqrt{6}}{6}-\frac{9\sqrt{6}}{4}+\frac{2\sqrt{6}}{9}\right)\div\frac{3\sqrt{6}}{2}\)
\(=-\frac{67\sqrt{6}}{36}\cdot\frac{2}{3\sqrt{6}}\)
\(=-\frac{67}{54}\)
\(\sqrt{\frac{5+2\sqrt{6}}{5-2\sqrt{6}}}+\sqrt{\frac{5-2\sqrt{6}}{5+2\sqrt{6}}}\)
\(=\sqrt{\frac{3+2\sqrt{3}\sqrt{2}+2}{3-2\sqrt{3}\sqrt{2}+2}}+\sqrt{\frac{3-2\sqrt{3}\sqrt{2}+2}{3+2\sqrt{3}\sqrt{2}+2}}\)
\(=\sqrt{\frac{\left(\sqrt{2}+\sqrt{3}\right)^2}{\left(\sqrt{2}-\sqrt{3}\right)^2}}+\sqrt{\frac{\left(\sqrt{2}-\sqrt{3}\right)^2}{\left(\sqrt{2}+\sqrt{3}\right)^2}}\)
\(=\frac{\left(\sqrt{2}+\sqrt{3}\right)}{\left(\sqrt{2}-\sqrt{3}\right)}+\frac{\left(\sqrt{2}-\sqrt{3}\right)}{\left(\sqrt{2}+\sqrt{3}\right)}\)\
\(=\frac{\left(\sqrt{2}+\sqrt{3}\right)\left(\sqrt{2}+\sqrt{3}\right)+\left(\sqrt{2}-\sqrt{3}\right)\left(\sqrt{2}-\sqrt{3}\right)}{\left(\sqrt{2}-\sqrt{3}\right)\left(\sqrt{2}+\sqrt{3}\right)}\)
\(=\frac{\left(\sqrt{2}+\sqrt{3}\right)^2+\left(\sqrt{2}-\sqrt{3}\right)^2}{\left(\sqrt{2}-\sqrt{3}\right)\left(\sqrt{2}+\sqrt{3}\right)}\)
\(=\frac{5+2\sqrt{6}+5-2\sqrt{6}}{\left(\sqrt{2}-\sqrt{3}\right)\left(\sqrt{2}+\sqrt{3}\right)}\)
\(=10\)
\(\frac{3+2\sqrt{3}}{\sqrt{3}}+\frac{2+\sqrt{2}}{\sqrt{2}+1}-\left(\sqrt{2}+3\right)\)
\(=\frac{\sqrt{3}\left(\sqrt{3}+2\right)}{\sqrt{3}}+\frac{\sqrt{2}\left(\sqrt{2}+1\right)}{\sqrt{2}+1}-\left(\sqrt{2}+3\right)\)
\(=\sqrt{3}+2+\sqrt{2}-\sqrt{2}-3\)
\(=\sqrt{3}-1\)