a: \(\frac{8^2\cdot4^5}{2^{20}}=\frac{\left(2^3\right)^2\cdot\left(2^2\right)^5}{2^{20}}\)
\(=\frac{2^6\cdot2^{10}}{2^{20}}=\frac{2^{16}}{2^{20}}=\frac{1}{2^4}=\frac{1}{16}\)
b: \(\frac{81^{11}\cdot3^{17}}{27^{10}\cdot9^{15}}\)
\(=\frac{\left(3^4\right)^{11}\cdot3^{17}}{\left(3^3\right)^{10}\cdot\left(3^2\right)^{15}}\)
\(=\frac{3^{44}\cdot3^{17}}{3^{30}\cdot3^{30}}=\frac{3^{61}}{3^{60}}=3\)
c: \(\frac{8^{10}+4^{10}}{8^4+4^{11}}\)
\(=\frac{\left(2^3\right)^{10}+\left(2^2\right)^{10}}{\left(2^3\right)^4+_{}\left(2^2\right)^{11}}\)
\(=\frac{2^{30}+2^{20}}{2^{12}+2^{22}}=\frac{2^{20}\left(2^{10}+1\right)}{2^{12}\left(2^{10}+1\right)}=2^8=256\)
