a) [latex]frac{2}{(2-sqrt{2})+sqrt{3}}=[/latex] [latex]frac{2[(2-sqrt{2})-sqrt{3}]}{[(2-sqrt{2})+sqrt{3}][(2-sqrt{2})-sqrt{3}]}=[/latex] [latex]frac{2[2-sqrt{2}-sqrt{3}]}{(2-sqrt{2})^2-(sqrt{3})^2}=[/latex] [latex]frac{2[2-sqrt{2}-sqrt{3}]}{4-4sqrt{2}+2-3}=[/latex] [latex]frac{2[2-sqrt{2}-sqrt{3}]}{3-4sqrt{2}}=[/latex] [latex]frac{(4-2sqrt{2}-2sqrt{3})(3+4sqrt{2})}{(3-4sqrt{2})(3+4sqrt{2})}=[/latex] [latex]frac{12+16sqrt{2}-6 sqrt{2}-16-6sqrt{3}-8sqrt{6}}{9-32}=[/latex] [latex]frac{-8sqrt{6}-6sqrt{3}+10sqrt{2}-4}{-23}=[/latex] [latex]frac{8sqrt{6}+6sqrt{3}-10sqrt{2}+4}{23}[/latex] ================== b) [latex]frac{2}{2-(sqrt{2}+sqrt{3})}=[/latex] [latex]frac{2[2+(sqrt{2}+sqrt{3})]}{[2-(sqrt{2}+sqrt{3})][2+(sqrt{2}+sqrt{3})]}=[/latex] [latex]frac{2[2+sqrt{2}+sqrt{3}]}{2^2-(sqrt{2}+sqrt{3})^2}=[/latex] [latex]frac{4+2sqrt{2}+2sqrt{3}}{4-(2+2 sqrt{6}+3)}=[/latex] [latex]frac{4+2sqrt{2}+2sqrt{3}}{4-2-2 sqrt{6}-3}=[/latex] [latex]frac{4+2sqrt{2}+2sqrt{3}}{-1-2 sqrt{6}}=[/latex] [latex]-frac{4+2sqrt{2}+2sqrt{3}}{1+2 sqrt{6}}=[/latex] [latex]-frac{(4+2sqrt{2}+2sqrt{3})(1-2sqrt{6})}{(1+2 sqrt{6})(1-2 sqrt{6})}=[/latex] [latex]-frac{4-8sqrt{6} +2sqrt{2}-4sqrt{12} +2sqrt{3}-4sqrt{18}}{1^2-(2 sqrt{6})^2}=[/latex] [latex]-frac{4-8sqrt{6} +2sqrt{2}-8sqrt{3} +2sqrt{3}- 12sqrt{2}}{1-24}=[/latex] [latex]-frac{-8sqrt{6}-6sqrt{3}-10sqrt{2}+4}{-23}=[/latex] [latex]frac{-8sqrt{6}-6sqrt{3}-10sqrt{2}+4}{23}[/latex] ================== c) [latex]frac{2}{1- sqrt[3]{3}+ sqrt[3]{9}}=[/latex] [latex]frac{2}{1- sqrt[3]{3}+ sqrt[3]{3^2}}=[/latex] [latex]frac{2}{1- sqrt[3]{3}+(sqrt[3]{3})^2}=[/latex] --------------------- Wzór: [latex]a^3+b^3=(a+b)(a^2-ab+b^2)[/latex] [latex]frac{1}{a^2-ab+b^2}= frac{a+b}{a^3+b^3}[/latex] Podstawiamy [latex]a=1[/latex] [latex]b=sqrt[3]{3}[/latex] --------------------- [latex]frac{2(1+sqrt[3]{3})}{1^2+(sqrt[3]{3})^3}=[/latex] [latex]frac{2(1+sqrt[3]{3})}{1+3}=[/latex] [latex]frac{2(1+sqrt[3]{3})}{4}=[/latex] [latex]frac{1+sqrt[3]{3}}{2}[/latex]
