[latex]frac{x^2}{y^3}=_{x=3sqrt{2}-1; y=sqrt{2}+1} = frac{(3sqrt{2}-1)^2}{(sqrt{2}+1)^3}=[/latex] [latex]= frac{(3sqrt{2})^2-2 cdot 3sqrt{2} cdot 1+1^2}{(sqrt{2})^3+3 cdot (sqrt{2})^2 cdot 1+3 cdot sqrt{2} cdot 1^2+1^3}= frac{18 - 6sqrt{2}+1}{2sqrt{2} + 6 + 3sqrt{2}+ 1}=[/latex] [latex]= frac{19 - 6sqrt{2}}{5sqrt{2} +7}= frac{(19 - 6sqrt{2})(5sqrt{2} - 7)}{(5sqrt{2} + 7)(5sqrt{2} - 7)}= frac{95sqrt{2}-133-60+42sqrt{2}}{50-49}= frac{137sqrt{2}-193}{1} =[/latex] [latex]=137sqrt{2}-193[/latex]
[latex]frac{x^2}{y^3}=frac{({3sqrt{2}-1)}^2}{{(sqrt{2}+1)}^3}=frac{3^2cdot {(sqrt2)}^2-2cdot3sqrt2+1}{(sqrt2)^3+3cdot(sqrt2)^2+3sqrt2+1}=frac{9cdot 2-6sqrt2+1}{2sqrt2+3cdot 2+3sqrt2+1}=\\frac{19-6sqrt2}{7+5sqrt2}=frac{(19-6sqrt2)(7-5sqrt2)}{(7+5sqrt2)(7-5sqrt2)}=frac{133-95sqrt2-42sqrt2+60}{7^2-5^2(sqrt2)^2}=\frac{193-137sqrt2}{49-50}=-(193-137sqrt2)=137sqrt2-193[/latex]
