a) [latex]frac{5^5}{10^4} = frac{5^5}{ (2 cdot 5)^4} = frac{5^5}{ 2^4 cdot 5^4} = frac{5^{5-4}}{ 2^4} = frac{5^1}{ 16} = frac{5}{ 16} [/latex] b) [latex]frac{44^4}{22^3} = frac{(2 cdot 22)^4}{ 22^3} = frac{2^4 cdot 22^4}{ 22^3} = 2^4 cdot 22^{4-3} = 16 cdot 22 = 352[/latex] c) [latex] frac{64^2 cdot 36^2}{6^3 cdot 2^7} = frac{(2^6)^2 cdot (6^2)^2}{ 6^3 cdot 2^7} = frac{2^{12} cdot 6^4}{ 6^3 cdot 2^7} = 2^{12-7} cdot 6^{4-3} =2^5 cdot 6^1= 32 cdot 6 = 192[/latex] d) [latex]frac{4^6 cdot 8^6}{32^5} = frac{(2^2)^6 cdot (2^3)^6}{ (2^5)^5} = frac{2^{12} cdot 2^{18}}{ 2^{25}} = frac{2^{12+18}}{ 2^{25}} =frac{2^{30}}{ 2^{25}} =2^{30-25} =2^5 = 32 [/latex] e) [latex]frac{(-3)^9 cdot 5^9}{15^8} = frac{(-3 cdot 5)^9}{15^8} = frac{(-15)^9}{15^8} =frac{(-1 cdot 15)^9}{15^8} =frac{(-1)^9 cdot 15^9}{15^8} =-1 cdot 15^{9-8} =[/latex] [latex]=-1 cdot 15^1 = -15 [/latex] f) [latex] frac{2^8 cdot 5^7}{0,1^4 cdot 100^6} = frac{2^8 cdot 5^7}{(frac{1}{10})^4 cdot (10^2)^6} = frac{2^8 cdot 5^7}{10^{-4} cdot 10^{12}} = frac{2^8 cdot 5^7}{10^{-4+12} } =frac{2^8 cdot 5^7}{10^8} =frac{2^8 cdot 5^7}{(2 cdot 5)^8} = [/latex] [latex]= frac{2^8 cdot 5^7}{2^8 cdot 5^8} = frac{2^{8-8}}{5^{8-7}} =frac{2^0}{5^1} =frac{1}{5}[/latex]
