2. a) [latex] frac{a^4*a^7}{a^{11}:a^5} = frac{a^{4+7}}{a^{11-5}} =frac{a^{11}}{a^6} = a^{11-6} = a^5 [/latex] b) [latex] frac{a^{15}:a^3}{a^4*a^6}= frac{a^{15-3}}{a^{4+6}} =frac{a^{12}}{a^{10}} = a^{12-10}=a^2 [/latex] c) [latex]frac{(a^5)^3}{a^8*a^4} = frac{a^{5*3}}{a^{8+4}} =frac{a^{15}}{a^{12}}=a^{15-12} = a^3 [/latex] d) [latex]frac{a^{16}:a^{4}}{(a^{5})^2}=frac{a^{16-4}}{a^{5*2}} = frac{a^{12}}{a^{10}} = a^{12-10}=a^2 [/latex] e) [latex](frac{a^{4}*(a^2)^3}{a^{5}} )^{10}=(frac{a^{4}*a^{2*3}}{a^{5}} )^{10}=(frac{a^{4+6}}{a^{5}} )^{10}=(frac{a^{10}}{a^{5}} )^{10}=(a^{5})^{10} = a^{5*10}= a^{50}[/latex] f) [latex][a^6*a^5:(a^4)^2]^7 = [a^{6+5}:a^{4*2}]^7 = (a^{11-8})^7 = a^{3*7}=a^{21}[/latex] g) [latex][(a^5)^3*a^4*a^6]^2 = [a^{5*3+4+6}]^2 = a^{25*2}=a^{50}[/latex] 4. a) [latex]2^7*5^6=2^{1+6}*5^6 = 2*2^6*5^6=2*(2*5)^6= 2*10^6 = 2 000 000[/latex] b) [latex]20^8:2^7 = 10^8*2^8:2^7 = 10^8*2^{8-7}=2*10^8 =200000000[/latex] c) [latex]5^4*2^6=5^4*2^4*2^2 = (5*2)^4*4=4*10^4=40000[/latex] d) [latex]30^5:3^4=(10*3)^5:3^4=10^5*3^5:3^4=10^5*3^{5-4} = 300000[/latex] e) [latex] frac{3^3*(3^2)^2}{18^5:2^5}= frac{3^3*3^{2*2}}{9^5*2^5:2^5} =frac{3^3*3^4}{(3^2)^5*1}= frac{3^7}{3^{10}}=3^{7-10}=3^{-3}=( frac{1}{3} )^3 = frac{1}{27} [/latex] f) [latex] frac{6^5*36}{2^6*3^6}= frac{6^5*6^2}{(2*3)^6}= frac{6^7}{6^6} = 6[/latex] g) [latex] frac{3^9*(5^3)^3}{(15^4)^2}= frac{3^9*5^9}{15^8}= frac{15^9}{15^8}=15 [/latex] h) [latex] frac{7^4* frac{1}{14^4} }{ ( frac{1}{2})^2 *frac{1}{2} } = frac{7^4 *14^{-4}}{2^{-2}*2^{-1} }= frac{7^4 *7^{-4}*2^{-4}}{2^{-3}} }= frac{1*2^{-4}}{2^{-3}} = 2^{-4-(-3)}= 2^{-1} = frac{1}{2} [/latex] 5. a) [latex] frac{27*32}{36^2}= frac{3^3*2^4}{(6^2)^2}=frac{3^3*2^3*2}{6^4}=frac{6^3*2^}{6^4}= frac{2}{6} = frac{1}{3} [/latex] b) [latex] frac{5*900}{3^4*25^2}= frac{5*30^2}{3^4*(5^2)^2}= frac{5* 3^2*2^2*5^2 }{5^4*3^4} = frac{5^3*3^2*2^2}{5^3*5*3^2*3^2} =frac{2^2}{5*3^2}= frac{4}{5*9} = frac{4}{45} [/latex] c) [latex] frac{2^8*5^7}{100}= frac{2^8*5^7}{4*25}=frac{2^8*5^7}{2^2*5^2}=2^6*5^5=2*2^5*5^5=2*10^5=200000[/latex] d) [latex] frac{12^5}{2^3*4^2*81} = frac{3^5*4^5}{2^3*(2^2)^2*3^4} =frac{3*3^4*(2^2)^5}{2^3*2^4*3^4}=frac{3*2^{10}}{2^7}= 3*2^3=3*8=24 [/latex] e) [latex]frac{15^4}{5^3*3^6}= frac{5^4*3^4}{5^3*3^6}= frac{5^3*5*3^{4}}{5^3*3^2*3^4}=frac{5}{3^2}=frac{5}{9}[/latex] f) [latex] frac{(3^2)^4*49^6}{21^3*7^5}= frac{3^8*(7^2)^6}{3^3*7^3*7^5}= frac{3^8*7^{12}}{3^3*7^8}=3^5*7^4=3*21^4= 583443[/latex] ktoś źle wydrukował działanie albo potęgi bo nie powinno być konieczne użycie kalkulatora 6. a) [latex]frac{3^4*9}{3^5}= frac{3^4*3^2}{3^5}=frac{3^6}{3^5} = 3[/latex] b) [latex]frac{(5^4)^2*25}{125*(5^2)^2}=frac{5^8*5^2}{5^3*5^4}= frac{5^{10}}{5^7}= 5^3 = 125 [/latex] c) [latex] frac{2*4*8*16*32}{64*128} = frac{2*2^2*2^3*2^4*2^5}{2^6*2^7}= frac{2^{15}}{2^{13}}=2^2 =4 [/latex] d) [latex] frac{81^2*27}{3^5*3*9} = frac{(3^4)^2*3^3}{3^5*3*3^2}= frac{3^8*3^3}{3^8}=3^3=27[/latex] e) i f) są przycięte, muszę domyślać się znaków, więc jeśli wyniki są inne to nie z mojej pomyłki e) [latex] frac{1024*2^2}{16^2*4^2}= frac{2^{10}*2^2}{(2^4)^2*(2^2)^2}= frac{2^{12}}{2^8*2^4}= frac{2^{12}}{2^{12}}=1 [/latex] f) [latex] frac{(6^4)^3:36}{6^5*6^3}= frac{6^{12}:6^2}{6^5*6^3}= frac{6^{10}}{6^8}=6^2 =36[/latex] g) [latex] frac{1}{4} *8*32^2* frac{1}{256}=2^{-2}*2^2*(2^5)^2*2^{-8}= 2^{-2+2+10+(-8)}=2^2=4 [/latex]
