a)
[tex]\dfrac{3^{21}}{9^{10}}=\dfrac{3^{21}}{(3^2)^{10}}=\dfrac{3^{21}}{3^{20}}=3^1=3[/tex]
[tex]\dfrac{5^{19}}{25^{10}}=\dfrac{5^{19}}{(5^2)^{10}}=\dfrac{5^{19}}{5^{20}}=\dfrac1{5^1}=\dfrac15[/tex]
[tex]\dfrac{2^5\cdot 3^4}{6^4}=\dfrac{2^5\cdot 3^4}{(2\cdot3)^4}=\dfrac{2^5\cdot 3^4}{2^4\cdot3^4}=\dfrac{2^5}{2^4}=2^1=2[/tex]
[tex]\dfrac{2^{17}-2^{16}}{2^{15}}=\dfrac{2^{16+1}-2^{16}}{2^{15}}=\dfrac{2^{16}\cdot2^1-2^{16}\cdot1}{2^{15}}=\dfrac{2^{16}\cdot(2-1)}{2^{15}}=2^1\cdot1=2[/tex]
b)
[tex]\dfrac{3^{20}}{3^{22}-3^{21}}=\dfrac{3^{20}}{3^{21+1}-3^{21}}=\dfrac{3^{20}}{3^{21}\cdot3-3^{21}}=\dfrac{3^{20}}{3^{21}\cdot(3-1)}}=\dfrac1{3^1\cdot2}=\dfrac16[/tex]
[tex]\dfrac{6^{13}}{3^{14}}\cdot\dfrac{5^{10}}{10^{12}}=\dfrac{(2\cdot3)^{13}\cdot5^{10}}{3^{14}\cdot(2\cdot5)^{12}}=\dfrac{2^{13}\cdot3^{13}\cdot5^{10}}{3^{14}\cdot2^{12}\cdot5^{12}}=\dfrac{2^{1}}{3^{1}\cdot5^{2}}=\dfrac{2}{3\cdot25}=\dfrac2{75}[/tex]
[tex]\dfrac{2^{19}\cdot15^3\cdot3^{15}}{6^{18}\cdot5^2}= \dfrac{2^{19}\cdot(5\cdot3)^3\cdot3^{15}}{(2\cdot3)^{18}\cdot5^2}= \dfrac{2^{19}\cdot5^3\cdot3^3\cdot3^{15}}{2^{18}\cdot3^{18}\cdot5^2}= \dfrac{2^{1}\cdot5^3\cdot3^{18}}{3^{18}\cdot5^2}=\\\\\\{}\qquad\qquad\qquad \qquad\qquad\qquad\qquad \qquad\qquad\qquad =\dfrac{2\cdot5^3}{5^2} =\dfrac{2\cdot5^1}{1}=10[/tex]
