a) ${15}^2:15={15}^2:{15}^1={15}^{2-1}={15}^1=15$

b) $\frac{4^{180}}{4^{177}}=4^{180-177}=4^3=64$

c) $\frac{5^8\cdot 5^9}{5^{15}}=\frac{5^{8+9}}{5^{15}}=\frac{5^{17}}{5^{15}}=5^{17-15}=5^2=25$

d) ${\left(-189\right)}^8:{189}^7={189}^8:{189}^7={189}^{8-7}={189}^1=189$

e) ${\left(-6\right)}^{13}\cdot \frac{1}{6^{11}}$ $={\left(\left(-1\right)\cdot 6\right)}^{13}:6^{11}=$   ${\left(-1\right)}^{13}\cdot 6^{13}:6^{11}=\left(-1\right)\cdot 6^{13-11}=\left(-1\right)\cdot 6^2=\left(-1\right)\cdot 36=-36$  

f)  $\frac{{12}^3\cdot {12}^7}{{12}^{10}}=\frac{{12}^{3+7}}{{12}^{10}}=\frac{{12}^{10}}{{12}^{10}}=1$

g) $\frac{5^{20}}{5^5\cdot 5^{12}\cdot 25}$ $=\frac{5^{20}}{5^{5+12}\cdot 5^2}$ $=\frac{5^{20}}{5^{17}\cdot 5^2}=$ $\frac{5^{20}}{5^{17+2}}=\frac{5^{20}}{5^{19}}=5^1=5$

h) ${0.3}^5:{0.3}^3={0.3}^{5-3}={0.3}^2=0.09$

i) $\frac{{\left(\frac{1}{2}\right)}^{13}\cdot {\left(\frac{1}{2}\right)}^3}{{\left(\frac{1}{2}\right)}^{14}}$ $=\frac{{\left(\frac{1}{2}\right)}^{13+3}}{{\left(\frac{1}{2}\right)}^{14}}=\frac{{\left(\frac{1}{2}\right)}^{16}}{{\left(\frac{1}{2}\right)}^{14}}=$ ${\left(\frac{1}{2}\right)}^{16-14}={\left(\frac{1}{2}\right)}^2=\frac{1}{4}$