\(\frac{45^{10}.5^{20}}{75^{15}}\)
\(=\frac{\left(15.3\right)^{10}.5^{20}}{\left(15.5\right)^{15}}\)
\(=\frac{15^{10}.3^{10}.5^{20}}{15^{15}.5^{15}}\)
\(=\frac{3^{10}.5^5}{15^5}=\frac{3^{10}.5^5}{3^5.5^5}=3^5=243\)
\(\frac{45^{10}.5^{20}}{75^{15}}=\frac{\left(9.5\right)^{10}.5^{20}}{\left(3.5.5\right)^{15}}=\frac{9^{10}.5^{10}.5^{20}}{3^{15}.5^{15}.5^{15}}=\frac{9^{10}.5^{30}}{3^{15}.5^{30}}=\frac{9^{10}}{3^{15}}=243\)