Sure, let's solve these equations:

1) \(\frac{2}{3} : \frac{1}{3} : 2x = -0.6\)

To solve:
- \(\frac{2}{3} / \frac{1}{3} = 2\)
- \(2x = -0.6 / 2\)
- \(x = -0.3\)

3) \(\frac{2x}{3} - \frac{1}{6} = \frac{x+4}{2}\)

To solve, find common denominator and isolate \(x\):
- Multiply through by 6: \(4x - 1 = 3(x + 4)\)
- \(4x - 1 = 3x + 12\)
- \(x = 13\)

5) \(\frac{1 - 5x}{2} = \frac{13}{36} = \frac{x+1}{3}\)

To solve:
- Use cross-multiplication and solve for \(x\):
- \(36(1 - 5x) = 13(x+1)\)
- \(36 - 180x = 13x + 13\)
- \(23 = 193x\)
- \(x = \frac{23}{193}\)

7) \(\left(\frac{27}{8}\right)^{\frac{9}{30}} \cdot x = \left(-\frac{3}{2}\right)^{\frac{20}{30}}\)

To solve:
- Simplify exponents:
- \((\frac{27}{8})^{3/10} \cdot x = (-\frac{3}{2})^{2/3}\)

11) \(\frac{5-x}{8} = \frac{x-1}{6}\)

To solve:
- Cross-multiply:
- \(6(5-x) = 8(x-1)\)
- \(30 - 6x = 8x - 8\)
- \(38 = 14x\)
- \(x = \frac{38}{14} = \frac{19}{7}\)

13) \(\frac{x-5}{-4} = \frac{-16}{x-5}\)

To solve:
- Cross-multiply and simplify:
- \((x-5)^2 = 64\)
- \(x-5 = \pm 8\)
- Solutions: \(x = 13\) or \(x = -3\)

15) \(2x^2 + (-6)^2 : 27 = 0\)

To solve:
- Simplify and solve for \(x\):
- \(2x^2 + 36 = 27\)
- \(2x^2 = -9\)
- No real solutions.

17) \(2^x + 2^{x+3} = 72 \, \text{for} \, x \in \mathbb{N}\)

To solve:
- \(2^x + 8 \cdot 2^x = 72\)
- \(9 \cdot 2^x = 72\)
- \(2^x = 8\)
- \(x = 3\)

Let me know if you need further explanations or solutions for specific problems!