To simplify each expression for \( A \), let's address them one by one:

1. **\( A = \sqrt{9} + 4\sqrt{2} - \sqrt{9} - 4\sqrt{2} \)**
\[
A = (\sqrt{9} - \sqrt{9}) + (4\sqrt{2} - 4\sqrt{2}) = 0 + 0 = 0
\]

2. **\( A = \sqrt{5} - 2\sqrt{6} + \sqrt{3} + 2\sqrt{2} \)**
This expression cannot be simplified further without additional context or specific simplification rules. It remains as:
\[
A = \sqrt{5} - 2\sqrt{6} + \sqrt{3} + 2\sqrt{2}
\]

3. **\( A = \sqrt{13} - 4\sqrt{3} + \sqrt{7} - 4\sqrt{3} \)**
\[
A = \sqrt{13} + \sqrt{7} - 8\sqrt{3}
\]

4. **\( A = \sqrt{5} + 2\sqrt{6} + \sqrt{5} - 2\sqrt{6} \)**
\[
A = ( \sqrt{5} + \sqrt{5} ) + ( 2\sqrt{6} - 2\sqrt{6} ) = 2\sqrt{5} + 0 = 2\sqrt{5}
\]

5. **\( A = \sqrt{24} + 8\sqrt{5} + \sqrt{9} - 4\sqrt{5} \)**
\[
\sqrt{24} = 2\sqrt{6}, \quad \sqrt{9} = 3
\]
\[
A = 2\sqrt{6} + 8\sqrt{5} + 3 - 4\sqrt{5} = 2\sqrt{6} + 4\sqrt{5} + 3
\]

6. **\( A = \sqrt{15} - 6\sqrt{6} + \sqrt{33} - 12\sqrt{6} \)**
\[
A = \sqrt{15} + \sqrt{33} - 18\sqrt{6}
\]

7. **\( A = \sqrt{6} - 4\sqrt{2} + \sqrt{22} - 12\sqrt{2} \)**
\[
A = \sqrt{6} + \sqrt{22} - 16\sqrt{2}
\]

8. **\( A = \sqrt{17} + 12\sqrt{2} + \sqrt{9} + 4\sqrt{2} \)**
\[
\sqrt{9} = 3
\]
\[
A = \sqrt{17} + 12\sqrt{2} + 3 + 4\sqrt{2} = \sqrt{17} + 16\sqrt{2} + 3
\]

These are the simplified forms of the given expressions, making use of basic arithmetic operations and properties of square roots. Each case is addressed according to its specific terms and radical expressions.