1.

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

3. \(A = \sqrt{20} - \sqrt{(\sqrt{5} - 1)^2} \cdot 2 - \frac{1}{\sqrt{5} + 2}\)
\[
\sqrt{20} = 2\sqrt{5} \\
\sqrt{(\sqrt{5} - 1)^2} = \sqrt{5} - 1 \\
A = 2\sqrt{5} - 2(\sqrt{5} - 1) - \frac{1}{\sqrt{5} + 2} = 2\sqrt{5} - 2\sqrt{5} + 2 - \frac{1}{\sqrt{5} + 2} = 2 - \frac{1}{\sqrt{5} + 2}
\]

 4. \(A = \frac{3 + 2\sqrt{3}}{\sqrt{3}} - \frac{1}{\sqrt{3} - \sqrt{2}} + \frac{2 + \sqrt{2}}{\sqrt{2} + 1}\)
\[
\frac{3 + 2\sqrt{3}}{\sqrt{3}} = \frac{3}{\sqrt{3}} + \frac{2\sqrt{3}}{\sqrt{3}} = \sqrt{3} + 2 \\
\frac{1}{\sqrt{3} - \sqrt{2}} = \frac{\sqrt{3} + \sqrt{2}}{(\sqrt{3})^2 - (\sqrt{2})^2} = \frac{\sqrt{3} + \sqrt{2}}{3 - 2} = \sqrt{3} + \sqrt{2} \\
\frac{2 + \sqrt{2}}{\sqrt{2} + 1} = \frac{(2 + \sqrt{2})(\sqrt{2} - 1)}{(\sqrt{2} + 1)(\sqrt{2} - 1)} = \frac{2\sqrt{2} - 2 + 2 - \sqrt{2}}{1} = \sqrt{2}
\]
\[
A = (\sqrt{3} + 2) - (\sqrt{3} + \sqrt{2}) + \sqrt{2} = 2
\]

5. \(A = \left(\frac{1}{\sqrt{3} - \sqrt{2}} - \frac{\sqrt{3} - \sqrt{6}}{1 - \sqrt{2}} - \sqrt{3}\right) (\sqrt{2} + \sqrt{3})\)
\[
\frac{1}{\sqrt{3} - \sqrt{2}} = \sqrt{3} + \sqrt{2} \\
\frac{\sqrt{3} - \sqrt{6}}{1 - \sqrt{2}} = \frac{(\sqrt{3} - \sqrt{6})(1 + \sqrt{2})}{(1 - \sqrt{2})(1 + \sqrt{2})} = \frac{\sqrt{3} + \sqrt{3}\sqrt{2} - \sqrt{6} - 2\sqrt{3}}{1 - 2} = \frac{\sqrt{3} + \sqrt{6} - \sqrt{6} - 2\sqrt{3}}{-1} = -\sqrt{3} - \sqrt{2}
\]
\[
A = (\sqrt{3} + \sqrt{2} - (-\sqrt{3} - \sqrt{2}) - \sqrt{3})(\sqrt{2} + \sqrt{3}) = (\sqrt{3} + \sqrt{2} + \sqrt{3} + \sqrt{2} - \sqrt{3})(\sqrt{2} + \sqrt{3}) = (\sqrt{3} + 2\sqrt{2})(\sqrt{2} + \sqrt{3})
\]

6. \(A = \left(5\sqrt{2} - \frac{5}{4}\sqrt{32 + \sqrt{200}}\right) \div \sqrt{8}\)
\[
\sqrt{32} = 4\sqrt{2}, \quad \sqrt{200} = 10\sqrt{2} \\
A = (5\sqrt{2} - \frac{5}{4}(4\sqrt{2} + \sqrt{10\sqrt{2}})) \div \sqrt{8} = (5\sqrt{2} - 5\sqrt{2} - \frac{5}{4}\sqrt{10\sqrt{2}}) \div \sqrt{8} = - \frac{5}{4}\sqrt{10\sqrt{2}} \div \sqrt{8} = -\frac{5}{4} \sqrt{\frac{10\sqrt{2}}{8}}
\]

7. \(A = 4\sqrt{5} - \sqrt{(3 - 2\sqrt{5})^2 \cdot 2 + \frac{1}{2}\sqrt{20} - \sqrt{45}}\)
\[
3 - 2\sqrt{5} = -\sqrt{5} \\
A = 4\sqrt{5} - \sqrt{(-\sqrt{5})^2 \cdot 2 + \frac{1}{2}\sqrt{20} - \sqrt{45}} = 4\sqrt{5} - \sqrt{5^2 \cdot 2 + \frac{1}{2} \cdot 2\sqrt{5} - 3\sqrt{5}} = 4\sqrt{5} - \sqrt{50}
\]

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