Решение:

1) $$\frac{(-5)^6 \cdot (-5)^7 \cdot (-5)^8}{(-5)^{14} \cdot (-5)^4} = \frac{(-5)^{6+7+8}}{(-5)^{14+4}} = \frac{(-5)^{21}}{(-5)^{18}} = (-5)^{21-18} = (-5)^3 = -125$$

2) $$\frac{1.2^{40} \cdot 1.2^{25} \cdot 1.2^4}{1.2^{59} \cdot 1.2^8} = \frac{1.2^{40+25+4}}{1.2^{59+8}} = \frac{1.2^{69}}{1.2^{67}} = 1.2^{69-67} = 1.2^2 = 1.44$$

3) $$\frac{(\frac{1}{3})^{10} \cdot (\frac{1}{3})^{20} \cdot (\frac{1}{3})^{30}}{(\frac{1}{3})^{34} \cdot (\frac{1}{3})^{23}} = \frac{(\frac{1}{3})^{10+20+30}}{(\frac{1}{3})^{34+23}} = \frac{(\frac{1}{3})^{60}}{(\frac{1}{3})^{57}} = (\frac{1}{3})^{60-57} = (\frac{1}{3})^3 = \frac{1}{27}$$

4) $$\frac{(-\frac{1}{6})^{25} \cdot (-\frac{1}{6})^{19} \cdot (-\frac{1}{6})^{16}}{(-\frac{1}{6})^{8} \cdot (-\frac{1}{6})^{49}} = \frac{(-\frac{1}{6})^{25+19+16}}{(-\frac{1}{6})^{8+49}} = \frac{(-\frac{1}{6})^{60}}{(-\frac{1}{6})^{57}} = (-\frac{1}{6})^{60-57} = (-\frac{1}{6})^3 = -\frac{1}{216}$$