a) \(2 \sqrt{\frac{2}{3}} - 4 \sqrt{\frac{3}{2}}\)

​​ \(= \frac{2 \sqrt{2}}{\sqrt{3}} - \frac{4 \sqrt{3}}{\sqrt{2}} = \frac{2 \sqrt{6}}{3} - 2 \sqrt{6} = - \frac{4 \sqrt{6}}{3}\)

b) \(\frac{5 \sqrt{48} - 3 \sqrt{27} + 2 \sqrt{12}}{\sqrt{3}}\)​

\(=\frac{5\cdot4\sqrt{3}-3\cdot3\sqrt{3}+2\cdot2\sqrt{3}}{\sqrt3}=\frac{\left(\right.20-9+4\left.\right)\sqrt{3}}{\sqrt3}=\frac{15\sqrt{3}}{\sqrt3}=15\)

c) \(\frac{1}{3 + 2 \sqrt{2}} + \frac{4 \sqrt{2} - 4}{2 - \sqrt{2}}\)

\(=\frac{3 - 2 \sqrt{2}}{\left(\right. 3 + 2 \sqrt{2} \left.\right) \left(\right. 3 - 2 \sqrt{2} \left.\right)}+\) \(\frac{\left(\right. 4 \sqrt{2} - 4 \left.\right) \left(\right. 2 + \sqrt{2} \left.\right)}{\left(\right. 2 - \sqrt{2} \left.\right) \left(\right. 2 + \sqrt{2} \left.\right)}=\)\(3-2\sqrt{2}+4\sqrt{2}=3+2\sqrt{2}=3+2\sqrt2\) \(\)

a) \(\left(\right. \sqrt{\frac{4}{3}} + \sqrt{3} \left.\right) . \sqrt{6}\).

\(=\left(\right.\frac{\sqrt{3}}{2}+\sqrt{3}\left.\right)\sqrt{6}=\frac{3 \sqrt{3}}{2}\cdot\sqrt{6}=\frac{3 \sqrt{18}}{2}=\frac{9 \sqrt{2}}{2}\)

b) \(\left(\right.1-2\sqrt{5}\left.\right)^2\).\(= 1 - 4 \sqrt{5} + 20 = 21 - 4 \sqrt{5}\)

c) \(2 \sqrt{3} - \sqrt{27}\). ​\(= 2 \sqrt{3} - 3 \sqrt{3} = - \sqrt{3}\)

d) \(\sqrt{45} - \sqrt{20} + \sqrt{5}\) .​ \(= 3 \sqrt{5} - 2 \sqrt{5} + \sqrt{5} = 2 \sqrt{5}\)

a)  \(A=\frac{\sqrt{3}}{\sqrt{\sqrt{3} + 1} - 1}-\frac{\sqrt{3}}{\sqrt{\sqrt{3} + 1} + 1}\).

\(A = \sqrt{3} \left(\right. \frac{1}{\sqrt{\sqrt{3} + 1} - 1} - \frac{1}{\sqrt{\sqrt{3} + 1} + 1} \left.\right)\) \(= \sqrt{3} \cdot \frac{2}{\left(\right. \sqrt{\sqrt{3} + 1} \left.\right)^{2} - 1} = \sqrt{3} \cdot \frac{2}{\sqrt{3}} = 2\)

\(A=2\)

b) \(B = \left(\right. \frac{15}{\sqrt{6} + 1} + \frac{4}{\sqrt{6} - 2} - \frac{12}{3 - \sqrt{6}} \left.\right) \left(\right. \sqrt{6} + 11 \left.\right)\).

\(B=\left[\right.3\left(\right.\sqrt{6}-1\left.\right)+2\left(\right.\sqrt{6}+2\left.\right)-4\left(\right.3+\sqrt{6}\left.\right)\left]\right.\left(\right.\sqrt{6}+11\left.\right)\) \(=\left(\sqrt{6}-11\left.\right)\left(\sqrt{6}+11\left.\right)\right.\right.\) \(=6-121B=-115\)\(\)

 \(c)C=4\sqrt{20}-3\sqrt{125}+5\sqrt{45}-15\sqrt{\frac{1}{5}}\).

\(C=8\sqrt{5}-15\sqrt{5}+15\sqrt{5}-3\sqrt{5}=5\sqrt{5}\) \(=5\sqrt{5}\)