[tex]Dana~~jest~~liczba:\\\\\frac{4+\sqrt{32} }{4} =\frac{4+\sqrt{16\cdot 2} }{4}=\frac{4+\sqrt{4^{2} \cdot 2} }{4}=\frac{4+4\sqrt{2} }{4}=\frac{4(1+\sqrt{2}) }{4}=1+\sqrt{2} \\\\Oznaczam~~dana~~liczbe:\\\\a=1+\sqrt{2} \\\\Wyznacze~~liczbe~~przeciwna~~~do~~a~~czyli~~-a\\\\Liczba~~przeciwna ~~wynosi:\\\\-(1+\sqrt{2} )=-1 -\sqrt{2} \\\\Wyznacze~~liczbe~~odwrotna~~do~~liczby~~a~~czyli~~ \frac{1}{a} \\\\pamietajac~~o~~usunieciu~~niewymiarnosci\\\\Liczba~~odwrotna ~~wynosi:[/tex]
[tex]\frac{1}{(1+\sqrt{2} )} \cdot \frac{(1-\sqrt{2} )}{(1-\sqrt{2} )} =\frac{1-\sqrt{2} }{1^{2} -(\sqrt{2} )^{2} } =\frac{1-\sqrt{2} }{1-2} = \frac{1-\sqrt{2} }{-1} =\frac{-1(\sqrt{2} -1)}{-1} =\sqrt{2} -1\\\\Aby~~usunac~~niewymiarnosc ~~z~~mianownika~~ulamka~~\\\\korzystam ~~ze~~wzoru~~skroconego ~~mnozenia:\\\\(x-y)\cdot(x+y)=x^{2} -y^{2}[/tex]
