zad1 a) [latex] 3^{-2} = ( frac{1}{3})^{2} = frac{1}{9} [/latex] b) [latex]5^{-2} = ( frac{1}{5})^{2} = frac{1}{25} [/latex] c) [latex]5^{-1} + 2^{-2}= ( frac{1}{5})^{1} + (frac{1}{2})^{2} = frac{1}{5} + frac{1}{4} = frac{4}{20}+ frac{5}{20}= frac{9}{20} [/latex] d) [latex]16^{ frac{3}{4}= sqrt[4]{16}^{3} = sqrt[4]{4096}= 8 [/latex] e) [latex]9^{ frac{1}{2}} = sqrt{9} =3[/latex] f) [latex]25^{ frac{1}{2}= sqrt{25}=5 [/latex] g) [latex]16^{- frac{1}{2}= frac{1}{ sqrt{16} }= frac{1}{4} [/latex] h) [latex]64^{- frac{1}{2}} = frac{1}{ sqrt{64} } = frac{1}{8} [/latex] i) [latex]27^{ frac{2}{3}}= sqrt[3]{27^2}= sqrt[3]{729}= 9 [/latex] j) [latex]64^{ frac{2}{3}} = sqrt[3]{64^2}= sqrt[3]{4096}= 16 [/latex] zad2 a) [latex]2^{2}*2^{4}= 2^{2} ^{+}^{4} = 2^{6} = 64 [/latex] b) [latex]3^{-5} * 3^{7} = 3^{-5 ^+ ^7}= 3^{2}=9 [/latex] c) [latex]6^{10} : 6^{8} = 6^ {10-8}= 6^{2} = 36[/latex] d) [latex](5^{2})^{3} : 5^{4} = 5^{2*3-4} = 5^{2} = 25[/latex] e) [latex] frac{2^4*2^{16} }{2^{17} } = 2^{4+16-17}= 2^{3} = 8[/latex] f) [latex] frac{(4 ^{-3}) ^{4}*4 ^{16} }{ 4^{6} } = 4^{-3*4+16-6}= 4^{-2} = (frac{1}{4})^{2} = frac{1}{16} [/latex] g) [latex] frac{ (5^{4} )^{3} *5^{6} }{5^{15}} = 5^{4*3+6-15} = 5^{3} =125[/latex] zad 3 a) [latex](3 + sqrt{5} )^{2} = 3^{2} + 2*3* sqrt{5} + (sqrt{5}) ^{2} = 9 + 6 sqrt{5}+5= 14 + 6 sqrt{5} [/latex] b) [latex] (sqrt{6} - 5) ( sqrt{6} +5)=( sqrt{6}) ^{2} - 5^{2} = 6-25=-19[/latex] c) [latex](5 - 2 sqrt{3}) ^{2}= 5^{2}-2*5* 2sqrt{3} + (2 sqrt{3}) ^{2} = 25 - 20 sqrt{3} +12=37- 20sqrt{3} [/latex] d) [latex]( sqrt{3} +4) ^{2} = (sqrt{5})^{2} + 2* sqrt{3}*4 + 4^{2} = 5+8 sqrt{3}+16=21+8 sqrt{3} [/latex] e)[latex]( sqrt{3}-4)( sqrt{3}+4) = (sqrt{3})^{ 2}-4^{2}=3-16=-13[/latex] f)[latex](3-2 sqrt{5})^{2}=3^{2} - 2*3*2 sqrt{5}+ (2 sqrt{5})^{2}= 9-12 sqrt{5}+20=29-12 sqrt{5} [/latex] zad4 g) [latex](1 + sqrt{2})^{2} = 1^{2} + 2*1* sqrt{2}+( sqrt{2})^{2} = 1+2 sqrt{2}+2=3+2 sqrt{2} [/latex] h) [latex]( sqrt{5}-3)( sqrt{5}+3)= (sqrt{5})^{2} - 3^{2} = 5-9=-4 [/latex] i)[latex](1+ sqrt{3})^{2} = 1^{2} + 2*1* sqrt{3} +( sqrt{3})^{2}= 1+2 sqrt{3}+3 = 4+2 sqrt{3} [/latex] j) [latex]( sqrt{2}-5)( sqrt{2}+5) = (sqrt{2})^{2} - 5^{2} = 2-25=-23 [/latex] LOGARYTMY wzory i własności logarytmów: [latex]log _{a}c=b <=> a^{b}=c [/latex] [latex]a^{log _{a}c}=c [/latex] [latex]log _{a}(x*y) = log_{a}x + log_{a}y [/latex] [latex]log_ax^{r} = r*log_ax[/latex] [latex]log_a frac{x}{y}=log_ax-log_ay [/latex] [latex]log _{3}81 = 4 [/latex] [latex]log_2 frac{1}{4}= -2 [/latex] [latex]log_ frac{1}{2}8= -3 [/latex] [latex]log _{12} frac{1}{12} =-1[/latex] [latex]log_ sqrt{7}}1=0 [/latex] [latex]log_{2}16=4[/latex] [latex]log100=2[/latex] [latex]log_{7}49=2[/latex] [latex]log_{3} frac{1}{9}=-2 [/latex] [latex]log_{2}32=5[/latex] [latex]log_{15} frac{1}{15}=-1 [/latex] [latex]log_{64}1=0[/latex] [latex]log1000=3[/latex] [latex]log_{4}16=2[/latex] [latex]log_{ frac{1}{5}}125=-3 [/latex] [latex]log_{8}64=2[/latex] a) [latex]log_{5}100-log_{5}4= log_5 frac{100}{4}= log_{5}25=2 [/latex] b) [latex]log_{2}80+log_{2}0,1=log_{2}(80*0,1)= log_2}8=3[/latex] c) [latex]3log_{40}2+log_{40}5=log_{40}(2^{3}*5) = log_{40} 40=1[/latex] d)[latex]log_{4}4-log_{4}64= log_{4} frac{4}{64}=log_{4} frac{1}{16}=-2 [/latex] e) [latex]log_{6}36-log_{3}27= 2-3=-1[/latex] f) [latex]log_{3}2-log_{3}18= log_{3} frac{2}{18} = log_{3} frac{1}{9}= -2 [/latex] a) [latex]log_{3}4,5+ log_{3}2= log_{3} (4,5*2)=log_{3}9=2[/latex] b) [latex]log_{5}75-log_{5}3= log_{5} frac{75}{3} =log_{5}25=2[/latex] c) [latex]2log_{36}3 +log{36}4= log_{36}(3^{2}*4)= log_{36}36=1[/latex] d)[latex]log_{3}27-log_{2}16= 3-4=-1[/latex] e) [latex]log_{5}5-log_{5}125= log_{5} frac{5}{125} = log_{5} frac{1}{25}=-2 [/latex] f)[latex]log_{4}2-log_{4}32= log_{4} frac{2}{32}= log_{4} frac{1}{16}= -2[/latex]
