$(6^4)^4 : (6^5)^3 = 6^{4*4} : 6^{5*3} = 6^{16} : 6^{15} = 6^{16 - 15} = 6^{1} = 6$

$8^3 : 4^4 = (2^3)^3 : (2^2)^4 = 2^{3 * 3} : 2^{2 * 4} = 2^{9} : 2^{8} = 2^{9 - 8} = 2^{1} = 2$

$\frac{7^{14} * (7^2)^3}{(7^3)^6 * 7^{2}} = \frac{7^{14} * 7^{2 * 3}}{7^{3 * 6} * 7^{2}} = \frac{7^{14} * 7^{6}}{7^{18} * 7^{2}} = \frac{7^{14 + 6}}{7^{18 + 2}} = \frac{7^{20}}{7^{20}} = 1$

$\frac{25^{3} * 125^2}{5^{10}} = \frac{(5^2)^3 * (5^3)^2}{5^{10}} = \frac{5^{2 * 3} * 5^{3 * 2}}{5^{10}} = \frac{5^{6} * 5^{6}}{5^{10}} = \frac{5^{6 + 6}}{5^{10}} = \frac{5^{12}}{5^{10}} = \frac{5^{2}}{1} = 25$

$\frac{3^{8} * 7^8}{21^{7}} = \frac{(3 * 7)^{8}}{21^{7}} = \frac{21^{8}}{21^{7}} = \frac{21^{1}}{1} = 21$

$\frac{5^{9} * 4^6}{20^{6}} = \frac{5^{9} * 4^6}{(5 * 4)^{6}} = \frac{5^{9} * 4^6}{5^{6} * 4^{6}} = \frac{5^{9}}{5^{6}} = \frac{5^{3}}{1} = 125$