`a) (45^10 . 5^10)/(75^10)`
`= (9^10. 5^10 . 5^10)/(3^10 . 25^10)`
`= (3^10 . 3^10. 5^20)/(3^10 . 5^20)`
`= 3^10`
`b) (2^17 . 9^4)/(6^3 . 8^3)`
`= (2^17 . 3^8)/(2^3 . 3^3 . 2^9)`
`= (2^12 . 2^5 . 3^3 . 3^5)/(2^12 . 3^3)`
`= 2^5 . 3^5`
`= 6^5`
`c) (8^10 + 4^10)/(8^4 + 4^11)`
`= (2^30 + 2^20)/(2^12 + 2^22)`
`= (2^30 + 2^20)/(2^12 + 2^22)`
`= [2^20 . (2^10 + 1)]/[2^12 . (1 + 2^10)]`
`= (2^20)/(2^12) `
`= 2^8`
\(a,\dfrac{45^{10}.5^{10}}{75^{10}}=\dfrac{\left(9.5\right)^{10}.5^{10}}{\left(3.25\right)^{10}}=\dfrac{9^{10}.5^{10}.5^{10}}{3^{10}.25^{10}}=\dfrac{\left(3^2\right)^{10}.5^{20}}{3^{10}.\left(5^2\right)^{10}}=\dfrac{3^{20}.5^{20}}{3^{10}.5^{20}}=3^{10}\)
\(b,\dfrac{2^{17}.9^4}{6^3.8^3}=\dfrac{2^{17}.\left(3^2\right)^4}{\left(2.3\right)^3.\left(2^3\right)^3}=\dfrac{2^{17}.3^8}{2^3.3^3.2^9}=\dfrac{2^{12}.2^5.3^3.3^5}{3^3.2^{12}}=2^5.3^5=\left(2.3\right)^5=6^5\)
\(c,\dfrac{8^{10}+4^{10}}{8^4+4^{11}}=\dfrac{\left(2^3\right)^{10}+\left(2^2\right)^{10}}{\left(2^3\right)^4+\left(2^2\right)^{11}}=\dfrac{2^{30}+2^{20}}{2^{12}+2^{22}}=\dfrac{\left(2^{10}+1\right).2^{20}}{\left(2^{10}+1\right).2^{12}}=\dfrac{2^{20}}{2^{12}}=2^8\)
