a: \(\frac{45^{10}\cdot5^{10}}{75^{10}}=\left(\frac{45\cdot5}{75}\right)^{10}=\left(\frac{225}{75}\right)^{10}=3^{10}\)
b: \(\frac{2^{17}\cdot9^4}{6^3\cdot8^3}=\frac{2^{17}\cdot3^8}{3^3\cdot2^3\cdot\left(2^3\right)^3}=\frac{2^{17}}{2^3\cdot2^9}\cdot3^5=2^5\cdot3^5=\left(2\cdot3\right)^5=6^5\)
c: \(\frac{8^{10}+4^{10}}{8^4+4^{11}}=\frac{2^{30}+2^{20}}{2^{12}+2^{22}}=\frac{2^{20}\left(2^{10}+1\right)}{2^{12}\left(2^{10}+1\right)}=\frac{2^{20}}{2^{12}}=2^8=256\)
