Oblicz: a) [latex]x^{3} - frac{1}{ x^{3} } [/latex] wiedząc że [latex] x- frac{1}{x} =2 [/latex] b)[latex] frac{ a^{2} +1}{a}[/latex] jeżeli [latex]sqrt{a} - frac{1}{ sqrt{a} } = 2[/latex]

Zauważ, że: [latex]left(x-frac{1}{x} ight)^{3}=x^{3}+frac{3}{x}-3x-left(frac{1}{x} ight)^{3}\ hbox{czyli:} \ left(x-frac{1}{x} ight)^{3}=x^{3}-frac{1}{x^{3}}-3(x-frac{1}{x}) \ hbox{Podstawiamy nasze} x-frac{1}{x}=2 : \ x^3-frac{1}{x^3}-3 cdot 2 = 2^{3} \ x^{3}-frac{1}{x^3}-6=8 \ oxed{x^{3}-frac{1}{x^{3}}=14}[/latex] W b) masz: [latex]left(sqrt{a}-frac{1}{sqrt{a}} ight)^{2}=a-2+frac{1}{a} \ hbox{Podstawiamy} sqrt{a}-frac{1}{sqrt{a}}=2: \ a-2+frac{1}{a}=2^{2} \ a+frac{1}{a}=4+2 \ frac{a^{2}}{a}+frac{1}{a}=6 \ oxed{frac{a^{2}+1}{a}=6}[/latex]