$1$. `{9^{15}.(-6)^{30}}/{27^{21}.8^{11}}`
`= {3^{30} . 2^{30} . 3^{30}}/{3^{63} . 2^{33}}`
`= {3^{60} . 2^{30}}/{3^{63} . 2^{33}}`
`= 1/{3^3.2^3}`
`= 1/{216}`
$2$. `{16^{11} . (-5)^{40}}/{(-10)^{41}}`
`= {2^{44} . 5^{40}}/{(-10)^{41}}`
`= {10^{40} . 2^4}/{(-10)^{41}}`
`= {16}/{-10}`
`= -8/5`
$3$. `{2^17.(-20)^{14}}/{125^5 . 24^7}`
`= {2^{17} . 4^{14} . 5^{14}}/{5^{15} . 3^7 . 8^7}`
`={2^{17} . 2^{28} . 5^{14}}/{5^{15} . 3^7 . 2^{21}}`
`= {2^{24}}/{5.3^7}`
$4$. ` {( -5)^{32}. 20^{43}}/{(-8)^{20} . 125^{25}}`
`= {5^{32} . 5^{43} . 4^{43}}/{2^{60} . 5^{75}}`
`= {5^{75} . 2^{86}}/{2^{60} . 5^{75}}`
`= 2^{26}`
$5$. `{27^2 . 25^3}/{6.15^4 .18}`
`= {3^6 . 5^6}/{2.3.3^4 . 5^4 . 3^2 . 2}`
`= {5^2}/{2^2.3}`
`= {25}/{12}`
$6$) `{20^3 . 49^2}/{14^3. (-5)^4}`
`= {4^3 . 5^3 . 7^4}/{2^3 . 7^3 . 5^4}`
`= {2^3. 7}/{5}`
`= {56}/5`