### 1. $A = \sqrt{21 - 4\sqrt{5}} - \sqrt{20}$

Rút gọn:

- Ta biết $\sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5}$.

Do đó, $A = \sqrt{21 - 4\sqrt{5}} - 2\sqrt{5}$.

### 2. $A = \sqrt{\left(\sqrt{5} + 1\right)^2 - \frac{5 - \sqrt{20}}{\sqrt{5}}}$

Rút gọn:

- $\left(\sqrt{5} + 1\right)^2 = 5 + 2\sqrt{5} + 1 = 6 + 2\sqrt{5}$.
- $\frac{5 - \sqrt{20}}{\sqrt{5}} = \frac{5 - 2\sqrt{5}}{\sqrt{5}} = \frac{5}{\sqrt{5}} - \frac{2\sqrt{5}}{\sqrt{5}} = \sqrt{5} - 2$.

Do đó, $A = \sqrt{6 + 2\sqrt{5} - (\sqrt{5} - 2)} = \sqrt{6 + 2\sqrt{5} - \sqrt{5} + 2} = \sqrt{8 + \sqrt{5}}$.

### 3. $A = 3\sqrt{8} - \sqrt{50} - \sqrt{\left(\sqrt{2} - 1\right)^2}$

Rút gọn:

- $3\sqrt{8} = 3 \cdot 2\sqrt{2} = 6\sqrt{2}$.
- $\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}$.
- $\sqrt{\left(\sqrt{2} - 1\right)^2} = |\sqrt{2} - 1| = \sqrt{2} - 1$.

Do đó, $A = 6\sqrt{2} - 5\sqrt{2} - (\sqrt{2} - 1) = 6\sqrt{2} - 5\sqrt{2} - \sqrt{2} + 1 = 1 + 1 = 1$.

### 4. $A = \sqrt{20 + \sqrt{180} - \sqrt{125}} : \sqrt{5}$

Rút gọn:

- $\sqrt{180} = \sqrt{36 \cdot 5} = 6\sqrt{5}$.
- $\sqrt{125} = \sqrt{25 \cdot 5} = 5\sqrt{5}$.

Do đó, $A = \sqrt{20 + 6\sqrt{5} - 5\sqrt{5}} : \sqrt{5} = \sqrt{20 + \sqrt{5}} : \sqrt{5} = \frac{\sqrt{20 + \sqrt{5}}}{\sqrt{5}} = \sqrt{\frac{20 + \sqrt{5}}{5}}$.

### 5. $A = 3\sqrt{3} - \sqrt{12} - \sqrt{\left(\sqrt{3} + 1\right)^2}$

Rút gọn:

- $\sqrt{12} = 2\sqrt{3}$.
- $\sqrt{\left(\sqrt{3} + 1\right)^2} = |\sqrt{3} + 1| = \sqrt{3} + 1$.

Do đó, $A = 3\sqrt{3} - 2\sqrt{3} - (\sqrt{3} + 1) = 3\sqrt{3} - 2\sqrt{3} - \sqrt{3} - 1 = -1$.

### 6. $A = (\sqrt{3} - \sqrt{75} + 4\sqrt{27}) \cdot \sqrt{3}$

Rút gọn:

- $\sqrt{75} = 5\sqrt{3}$.
- $4\sqrt{27} = 4 \cdot 3\sqrt{3} = 12\sqrt{3}$.

Do đó, $A = (\sqrt{3} - 5\sqrt{3} + 12\sqrt{3}) \cdot \sqrt{3} = (1\sqrt{3} - 5\sqrt{3} + 12\sqrt{3}) \cdot \sqrt{3} = 8\sqrt{3} \cdot \sqrt{3} = 24$.

### 7. $A = (\sqrt{20 + \sqrt{12}})(\sqrt{5} - \sqrt{3}) + 2\sqrt{3}$

Rút gọn:

- $\sqrt{12} = 2\sqrt{3}$.

Do đó, $A = (\sqrt{20 + 2\sqrt{3}})(\sqrt{5} - \sqrt{3}) + 2\sqrt{3}$.

### 8. $A = 4 + 2\sqrt{3} + \sqrt{7 - 4\sqrt{3}}$

Không cần rút gọn thêm.

### 9. $A = \sqrt{12 - \sqrt{27} + \frac{2}{\sqrt{3} - 1}}$

Rút gọn:

- $\sqrt{27} = 3\sqrt{3}$.
- $\frac{2}{\sqrt{3} - 1} = \frac{2}{\sqrt{3} - 1} \cdot \frac{\sqrt{3} + 1}{\sqrt{3} + 1} = \frac{2(\sqrt{3} + 1)}{3 - 1} = \sqrt{3} + 1$.

Do đó, $A = \sqrt{12 - 3\sqrt{3} + (\sqrt{3} + 1)} = \sqrt{12 - 2\sqrt{3} + 1} = \sqrt{13 - 2\sqrt{3}}$.

### 10. $A = \sqrt{9 - 4\sqrt{5} - \sqrt{5}}$

Không cần rút gọn thêm.