a)
[tex] \frac{ 6^{10} }{ 81^{-2}* 32^{-2} } =\\=6^{10} * 81^{2}* 32^{2} =\\=(2*3)^{10} * ( 3^{4}) ^{2}* (2^{5})^{2} =\\=2^{10}*3^{10} * 3^{8} * 2^{10}=\\=2^{20}*3^{18} [/tex]
b)
[tex] 20^{-4}* \frac{15^{-3}}{30^{-7}} =\\= \frac{1}{20^{4}}* \frac{30^{7}}{15^{3}} =\\=
\frac{1}{(2^{2}*5)^{4}}* \frac{(5*3*2)^{7}}{(3*5)^{3}} =\\=
\frac{1}{2^{8}*5^{4}}* \frac{5^{7}*3^{7}*2^{7}}{3^{3}*5^{3}} =\\=
\frac{5^{7}*3^{7}*2^{7}}{2^{8}*5^{4}*3^{3}*5^{3}} =\\=
\frac{5^{7}*3^{7}*2^{7}}{2^{8}*3^{3}*5^{7}} =\\= \frac{ 3^{4} }{2} [/tex]
