Bài 62. Rút gọn các biểu thức sau:
a) \(\frac{1}{2}\sqrt{48}-2\sqrt{75}-\frac{\sqrt{33}}{\sqrt{11}}+5\sqrt{1\frac{1}{3}}\);
b) \(\sqrt{150}+\sqrt{1,6}\cdot \sqrt{60}+4,5\cdot \sqrt{2\frac{2}{3}}-\sqrt{6};\)
c) \((\sqrt{28}-2\sqrt{3}+\sqrt{7})\sqrt{7}+\sqrt{48};\)
d) \((\sqrt{6}+\sqrt{5})^{2}-\sqrt{120}.\)
Hướng dẫn giải:
a) \(\frac{1}{2}\sqrt{48}-2\sqrt{75}-\frac{\sqrt{33}}{\sqrt{11}}+5\sqrt{1\frac{1}{3}}\)
\(=\frac{1}{2}\sqrt{16\cdot 3}-2\sqrt{25\cdot 3}-\sqrt{\frac{33}{11}}+5\sqrt{\frac{4}{3}}\)
\(=\frac{1}{2}\cdot 4\sqrt{3}-2\cdot 5\sqrt{3}-\sqrt{3}+5\cdot \frac{2}{3}\sqrt{3}\)
\(=(2-10-1+\frac{10}{3})\sqrt{3}\)
\(=-\frac{17}{3}\sqrt{3}.\)
b) \(\sqrt{150}+\sqrt{1,6}\cdot \sqrt{60}+4,5\cdot \sqrt{2\frac{2}{3}}-\sqrt{6}\)
\(=\sqrt{25\cdot 6}+\sqrt{1,6\cdot 60}+4,5\cdot \sqrt{\frac{8}{3}}-\sqrt{6}\)
\(= 5\sqrt{6}+\sqrt{16\cdot 6}+4,5\cdot \frac{\sqrt{8\cdot 3}}{3}-\sqrt{6}\)
\(=5\sqrt{6}+4\sqrt{6}+4,5\cdot 2\cdot \frac{\sqrt{6}}{3}-\sqrt{6}\)
\(=(5+4+3-1)\sqrt{6}=11\sqrt{6}.\)
c) \(=(\sqrt{28}-2\sqrt{3}+\sqrt{7})\sqrt{7}+\sqrt{84}\)
\(=(\sqrt{4.7}-2\sqrt{3}+\sqrt{7})\sqrt{7}+\sqrt{4.21}\)
\(= (2\sqrt{7}-2\sqrt{3}+\sqrt{7})\sqrt{7}+2\sqrt{21}\)
\(=2.7-2\sqrt{21}+7+2\sqrt{21}=21.\)
d) \((\sqrt{6}+\sqrt{5})^{2}-\sqrt{120}\)
\(=6+2\sqrt{6.5}+5-\sqrt{4.30}\)
\(=6+2\sqrt{30}+5-2\sqrt{30}=11.\)