${\Large\frac{6,57 · 10^{-2}}{10^{-4}}}=6,57 \cdot 10^{-2-(-4)}=6,57 \cdot 10^2=657$

$2^{16}·5^{17} ∶10^{15}={\Large\frac{2^{16}·5^{17}}{(2 \cdot 5)^{15}}} ={\Large\frac{2^{16}·5^{17}}{2^{15}·5^{15} }}=2 \cdot 5^2=2 \cdot 25=50$

${\Large\frac{log_9⁡20}{log_{81}⁡20}} ={\Large\frac{log_9⁡20}{log_{9^2}20}} ={\Large\frac{log_9⁡20}{\frac{1}{2} log_9⁡20}}=1∶{\Large\frac{1}{2}}=2$

$log_a⁡b+log_a⁡c=log_a⁡(bc)$

$log_3⁡1{,}2+log_3⁡22{,}5=log_3⁡(1{,}2 \cdot 22{,}5)=log_3⁡27=3$

$(ab)^m=a^m b^m,(\sqrt a)^2=a$

${\Large\frac{(9 · \sqrt{17})^2}{153}}={\Large\frac{9^2 \cdot(\sqrt{17})^2}{153}}={\Large\frac{81 \cdot 17}{153}}={\Large\frac{81 \cdot1}{9}}=9$

$(a^m )^n=a^{mn}, a^m ∶a^n=a^{m-n}$

${\Large\frac{(16^2)^2 · 49^{-7}}{7^{-15} \cdot4^7}}={\Large\frac{16^4 \cdot (7^2 )^{-7}}{7^{-15} \cdot 4^7}}={\Large\frac{(4^2 )^4 \cdot7^{-14}}{7^{-15} \cdot4^7}}={\Large\frac{4^8 \cdot 7^{15}}{4^7 \cdot 7^{14}}}=4 \cdot 7=28$

$a^m \cdot a^n=a^{m+n},a^m ∶a^n=a^{m-n}$

${\Large\frac{y^{-9} \cdot y^{-7}}{y^{-16}}} =y^{-9+(-7)-(-16)}=y^{-16+16}=y^0=1$

$log_a⁡ a^b =b$

$5^{5 - log_5⁡625}=5^{5 - log_5⁡ 5^4}=5^{5 - 4}=5^1=5$

$log_a⁡ b^c =c ~log_a⁡b, log_a⁡b-log_a⁡c=log_a⁡ {\Large\frac{b}{c}}$

$4log_5⁡ 9 - log_5⁡{\Large\frac{6561}{125}}=log_5⁡ 9^4 -log_5⁡{\Large\frac{6561}{125}}=log_5⁡{\Large\frac{6561 \cdot125}{6561}}=log_5⁡125$

$log_a⁡b^c=c~ log_a⁡b,log_a⁡b+log_a⁡c=log_a⁡(bc)$

$3·log_3⁡ 0{,}2 + log_3⁡ 1125=log_3⁡ 0{,}2^3 +log_3⁡ 1125=log_3⁡(0,008 \cdot 1125)=log_3⁡9=2$