8. [latex]a=left(frac{1}{6} ight)^{-2}+12^0-125^{frac{2}{3}}[/latex] [latex]a=6^2+1-(5^3)^{frac{2}{3}}[/latex] [latex]a=36+1-5^2[/latex] [latex]a=37-25[/latex] [latex]a=12[/latex] [latex]4 \% x=12[/latex] [latex]0,04x=12 /:0,04[/latex] [latex]x=300[/latex] =============== 9. [latex]W=(2- sqrt{2} )^3+(5-2 sqrt{2} )(5+2 sqrt{2} )-2(2 sqrt{2}-3)^2[/latex] [latex]W=2^3-3cdot 2^2 sqrt{2}+3cdot 2cdot (sqrt{2})^2+(sqrt{2})^3+[/latex] [latex]+5^2-(2 sqrt{2} )^2-2cdot [(2 sqrt{2})^2-2cdot 2 sqrt{2}cdot 3 +3^2][/latex] [latex]W=8-12sqrt{2}+12+2sqrt{2}++25-8-2cdot [8-12sqrt{2}+9][/latex] [latex]W=-14sqrt{2}+37-16+24sqrt{2}-18[/latex] [latex]W=10sqrt{2}+3[/latex] -------------------------- [latex]W-12 sqrt{2}=(3+2 sqrt{2} )^{-1}[/latex] [latex]L=W-12 sqrt{2}=10sqrt{2}+3-12 sqrt{2}=3-2 sqrt{2}=[/latex] [latex]frac{(3-2 sqrt{2})(3+2 sqrt{2})}{3+2 sqrt{2}}= frac{3^2-(2 sqrt{2})^2}{3+2 sqrt{2}}=[/latex] [latex]frac{9-8}{3+2 sqrt{2}}= frac{1}{3+2 sqrt{2}}=(3+2 sqrt{2})^{-1}=P[/latex] =============== 1. [latex]a=sqrt{14+6 sqrt{5}}-sqrt{5}[/latex] [latex]a=sqrt{5+6sqrt{5}+9}-sqrt{5}[/latex] [latex]a=sqrt{( sqrt{5} )^2+2cdot 3sqrt{5}+3^2}-sqrt{5}[/latex] [latex]a=sqrt{(sqrt{5} +3)^2}-sqrt{5}[/latex] [latex]a=|sqrt{5} +3|-sqrt{5}[/latex] [latex]a=sqrt{5} +3-sqrt{5}[/latex] [latex]a=3[/latex] =============== 2. [latex]x=m^2-m[/latex] Dla [latex]m=5 Rightarrow x=5^2-5=25-5=20[/latex] 20 nie jest liczbą podzielną przez 6 Twierdzenie jest fałszywe. =============== 3. [latex]frac{2^{-3}-left( frac{5}{4} ight)^{-2}cdot left( frac{1}{8} ight)^2 }{10^{-1}+left( - frac{1}{8} ight)^0 }=[/latex] [latex]frac{left( frac{1}{2} ight)^{3}-left( frac{4}{5} ight)^{2}cdot frac{1}{64}}{ frac{1}{10} +1}=[/latex] [latex]frac{frac{1}{8}-frac{16}{25}cdot frac{1}{64}}{1,1}=[/latex] [latex]frac{0,115}{1,1}= frac{115}{1100}= frac{23}{220}[/latex] =============== 4. a) [latex]sqrt{9-2 sqrt{14}} + sqrt{9+2 sqrt{14}}=[/latex] [latex]sqrt{7-2 sqrt{14}+2} + sqrt{7+2 sqrt{14}+2}=[/latex] [latex]sqrt{( sqrt{7} )^2-2 sqrt{7cdot 2}+( sqrt{2})^2 } + sqrt{( sqrt{7} )^2+2 sqrt{7cdot 2}+( sqrt{2})^2 }=[/latex] [latex]sqrt{( sqrt{7}-sqrt{2})^2 } + sqrt{( sqrt{7} + sqrt{2})^2 }=[/latex] [latex]|sqrt{7}-sqrt{2}|+ |sqrt{7} + sqrt{2}|=[/latex] [latex]sqrt{7}-sqrt{2}+sqrt{7} + sqrt{2}=2 sqrt{7}[/latex] Dana liczba nie jest liczbą naturalną -------------------------- b) [latex]left[ (2+ sqrt{3} )^3+(2- sqrt{3})^3 ight]^2=[/latex] [latex]left[2^3+3cdot2^2cdot sqrt{3}+3cdot 2cdot ( sqrt{3} )^2+ (sqrt{3})^3+[/latex] [latex]+2^3-3cdot2^2cdot sqrt{3}+3cdot 2cdot ( sqrt{3} )^2- (sqrt{3})^3 ight]^2=[/latex] [latex]left[8+18+ 8+18 ight]^2=52^2=2704[/latex] Dana liczba jest liczbą naturalną.
