\(a.\dfrac{15\cdot4^{12}\cdot9^7-4\cdot3^{15}\cdot8^8}{19\cdot2^{24}\cdot3^{14}-6\cdot4^{12}\cdot27^5}\\ =\dfrac{15\cdot\left(2^2\right)^{12}\cdot\left(3^2\right)^7-4\cdot3^{15}\cdot\left(2^3\right)^8}{19\cdot2^{24}\cdot3^{14}-6\cdot\left(2^2\right)^{12}\cdot\left(3^3\right)^5}\\ =\dfrac{15\cdot2^{24}\cdot3^{14}-4\cdot3^{15}\cdot2^{24}}{19\cdot2^{24}\cdot3^{14}-6\cdot2^{24}\cdot3^{15}}\\ =\dfrac{2^{24}\cdot3^{14}\cdot\left(15-4\cdot3\right)}{2^{24}\cdot3^{14}\cdot\left(19-6\cdot3\right)}\\ =\dfrac{15-12}{19-18}\\ =3\)
\(b.\dfrac{3^{15}\cdot2^{22}+6^{16}\cdot4^4}{2\cdot9^9\cdot8^7-7\cdot27^5\cdot2^{23}}\\ =\dfrac{3^{15}\cdot2^{22}+2^{16}\cdot3^{16}\cdot\left(2^2\right)^4}{2\cdot\left(3^2\right)^9\cdot\left(2^3\right)^7-7\cdot\left(3^3\right)^5\cdot2^{23}}\\ =\dfrac{3^{15}\cdot2^{22}+2^{24}\cdot3^{16}}{3^{18}\cdot2^{22}-7\cdot3^{15}\cdot2^{23}}\\ =\dfrac{3^{15}\cdot2^{22}\cdot\left(1+2^2\cdot3\right)}{3^{15}\cdot2^{22}\cdot\left(3^3-7\cdot2\right)}\\ =\dfrac{1+12}{27-14}\\ =1\)
\(c.\dfrac{16^3\cdot3^{10}+120\cdot6^9}{4^6\cdot3^{12}+6^{11}}\\ =\dfrac{\left(2^4\right)^3\cdot3^{10}+2^3\cdot3\cdot5\cdot2^9\cdot3^9}{\left(2^2\right)^6\cdot3^{12}+2^{11}\cdot3^{11}}\\ =\dfrac{2^{12}\cdot3^{10}+2^{12}\cdot3^{10}\cdot5}{2^{12}\cdot3^{12}+2^{11}\cdot3^{11}}\\ =\dfrac{2^{12}\cdot3^{10}\cdot\left(1+5\right)}{2^{11}\cdot3^{11}\cdot\left(2\cdot3+1\right)}\\ =\dfrac{1+5}{6+1}\\ =\dfrac{6}{7}\)