Let's simplify each expression step by step:

1. **\(2\sqrt{75} - 5\sqrt{27} - \sqrt{192} + 4\sqrt{48}\)**:
- \(\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}\)
- \(\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}\)
- \(\sqrt{192} = \sqrt{64 \times 3} = 8\sqrt{3}\)
- \(\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}\)

Plug back in:
- \(2\cdot 5\sqrt{3} - 5\cdot 3\sqrt{3} - 8\sqrt{3} + 4\cdot 4\sqrt{3}\)
- \(10\sqrt{3} - 15\sqrt{3} - 8\sqrt{3} + 16\sqrt{3}\)
- \((10 - 15 - 8 + 16)\sqrt{3} = 3\sqrt{3}\)

2. **\(2\sqrt{48} + 4\sqrt{27} + \sqrt{75} + 2\sqrt{3}\)**:
- \(\sqrt{48} = 4\sqrt{3}\)
- \(\sqrt{27} = 3\sqrt{3}\)
- \(\sqrt{75} = 5\sqrt{3}\)

Plug back in:
- \(2 \cdot 4\sqrt{3} + 4 \cdot 3\sqrt{3} + 5\sqrt{3} + 2\sqrt{3}\)
- \(8\sqrt{3} + 12\sqrt{3} + 5\sqrt{3} + 2\sqrt{3}\)
- \((8 + 12 + 5 + 2)\sqrt{3} = 27\sqrt{3}\)

3. **\(5\sqrt{12} + 3\sqrt{27} - 2\sqrt{108} - \sqrt{192}\)**:
- \(\sqrt{12} = 2\sqrt{3}\)
- \(\sqrt{27} = 3\sqrt{3}\)
- \(\sqrt{108} = 6\sqrt{3}\)
- \(\sqrt{192} = 8\sqrt{3}\)

Plug back in:
- \(5\cdot 2\sqrt{3} + 3\cdot 3\sqrt{3} - 2\cdot 6\sqrt{3} - 8\sqrt{3}\)
- \(10\sqrt{3} + 9\sqrt{3} - 12\sqrt{3} - 8\sqrt{3}\)
- \((10 + 9 - 12 - 8)\sqrt{3} = -1\sqrt{3}\)

4. **\(4\sqrt{12} + \sqrt{108} - 8\sqrt{3} + \sqrt{7} - 7\sqrt{3}\)**:
- \(\sqrt{12} = 2\sqrt{3}\)
- \(\sqrt{108} = 6\sqrt{3}\)

Plug back in:
- \(4\cdot 2\sqrt{3} + 6\sqrt{3} - 8\sqrt{3} + \sqrt{7} - 7\sqrt{3}\)
- \(8\sqrt{3} + 6\sqrt{3} - 8\sqrt{3} - 7\sqrt{3} + \sqrt{7}\)
- \((8 + 6 - 8 - 7)\sqrt{3} + \sqrt{7} = -1\sqrt{3} + \sqrt{7}\)

5. **\(3\sqrt{45} - 7\sqrt{125} + \sqrt{500} + 16\sqrt{9} - 4\sqrt{5}\)**:
- \(\sqrt{45} = 3\sqrt{5}\)
- \(\sqrt{125} = 5\sqrt{5}\)
- \(\sqrt{500} = 10\sqrt{5}\)
- \(\sqrt{9} = 3\)

Plug back in:
- \(3\cdot 3\sqrt{5} - 7\cdot 5\sqrt{5} + 10\sqrt{5} + 16\cdot 3 - 4\sqrt{5}\)
- \(9\sqrt{5} - 35\sqrt{5} + 10\sqrt{5} + 48 - 4\sqrt{5}\)
- \((9 - 35 + 10 - 4)\sqrt{5} + 48\)
- \((-20\sqrt{5} + 48)\)

6. **\(5\sqrt{18} - 3\sqrt{32} + \frac{1}{5}\sqrt{50}\)**:
- \(\sqrt{18} = 3\sqrt{2}\)
- \(\sqrt{32} = 4\sqrt{2}\)
- \(\sqrt{50} = 5\sqrt{2}\)

Plug back in:
- \(5\cdot 3\sqrt{2} - 3\cdot 4\sqrt{2} + \frac{1}{5}\cdot 5\sqrt{2}\)
- \(15\sqrt{2} - 12\sqrt{2} + \sqrt{2}\)
- \((15 - 12 + 1)\sqrt{2} = 4\sqrt{2}\)

These simplified expressions now retain their combined square root forms:
1. \(3\sqrt{3}\)
2. \(27\sqrt{3}\)
3. \(-\sqrt{3}\)
4. \(-\sqrt{3} + \sqrt{7}\)
5. \(-20\sqrt{5} + 48\)
6. \(4\sqrt{2}\)