Câu 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) \(\sqrt{150}+\sqrt{1,6} \cdot \sqrt{60}+4,5 \sqrt{2 \frac{2}{3}}-\sqrt{6}\)

c) \((\sqrt{28}-2 \sqrt{3}+\sqrt{7}) \sqrt{7}+\sqrt{84}\)

d) \((\sqrt{6}+\sqrt{5})^{2}-\sqrt{120}\)

Lời giải:

a) \(\frac{1}{2} \sqrt{48}-2 \sqrt{75}-\frac{\sqrt{33}}{\sqrt{11}}+5 \sqrt{1 \frac{1}{3}}\)

\(= \frac{1}{2} \sqrt{16.3}-2.5 \sqrt{3}-\sqrt{3}-\frac{10}{3} \sqrt{3}\)

\(= 2 \sqrt{3}-10 \sqrt{3}-\sqrt{3}-\frac{10}{3} \sqrt{3}\)

\(=-9 \sqrt{3}+\frac{10}{3} \sqrt{3}=\left(-9+\frac{10}{3}\right) \sqrt{3}=-\frac{17}{3} \sqrt{3}\)

b) \(\sqrt{150}+\sqrt{1,6} \cdot \sqrt{60}+4,5 \cdot \sqrt{2 \frac{2}{3}}-\sqrt{6}\)

\(=\sqrt{25.6}+\sqrt{1,6.60}+4,5 \cdot \sqrt{\frac{2.3+2}{3}}-\sqrt{6}\)

\(=\sqrt{5^{2} \cdot 6}+\sqrt{1,6 \cdot(6 \cdot 10)}+4,5 \sqrt{\frac{8}{3}}-\sqrt{6}\)

\(=5 \sqrt{6}+\sqrt{(1,6 \cdot 10) \cdot 6}+4,5 \frac{\sqrt{8}}{\sqrt{3}}-\sqrt{6}\)

\(=5 \sqrt{6}+\sqrt{16.6}+4,5 \frac{\sqrt{8} \cdot \sqrt{3}}{3}-\sqrt{6}\)

\(=5 \sqrt{6}+\sqrt{4^{2} \cdot 6}+4,5 \frac{\sqrt{8.3}}{3}-\sqrt{6}\)

\(=5 \sqrt{6}+4 \sqrt{6}+4,5 \cdot \frac{\sqrt{4.2 .3}}{3}-\sqrt{6}\)

\(=5 \sqrt{6}+4 \sqrt{6}+4,5 \cdot \frac{\sqrt{2^{2} \cdot 6}}{3}-\sqrt{6}\)

\(=5 \sqrt{6}+4 \sqrt{6}+4,5.2 \frac{\sqrt{6}}{3}-\sqrt{6}\)

\(=5 \sqrt{6}+4 \sqrt{6}+9 \frac{\sqrt{6}}{3}-\sqrt{6}\)

\(=5 \sqrt{6}+4 \sqrt{6}+3 \sqrt{6}-\sqrt{6}\)

\(=(5+4+3-1) \sqrt{6}=11 \sqrt{6}\)

c) \((\sqrt{28}-2 \sqrt{3}+\sqrt{7}) \sqrt{7}+\sqrt{84}\)

\(=(\sqrt{4.7}-2 \sqrt{3}+\sqrt{7}) \sqrt{7}+\sqrt{4.21}\)

\(=\left(\sqrt{2^{2} .7}-2 \sqrt{3}+\sqrt{7}\right) \sqrt{7}+\sqrt{2^{2} \cdot 21}\)

\(=(2 \sqrt{7}-2 \sqrt{3}+\sqrt{7}) \sqrt{7}+2 \sqrt{21}\)

\(=2 \sqrt{7} \cdot \sqrt{7}-2 \sqrt{3} \cdot \sqrt{7}+\sqrt{7} \cdot \sqrt{7}+2 \sqrt{21}\)

\(=2 \cdot(\sqrt{7})^{2}-2 \sqrt{3.7}+(\sqrt{7})^{2}+2 \sqrt{21}\)

\(=2.7-2 \sqrt{21}+7+2 \sqrt{21}\)

\(=14-2 \sqrt{21}+7+2 \sqrt{21}\)

\(=14+7=21\)

d) \((\sqrt{6}+\sqrt{5})^{2}-\sqrt{120}\)

\(=(\sqrt{6})^{2}+2 \sqrt{6} \cdot \sqrt{5}+(\sqrt{5})^{2}-\sqrt{4.30}\)

\(= 6+2 \sqrt{30}+5-2 \sqrt{30}=11\)