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Fie numerele a=[tex] \sqrt{9-2 \sqrt{14} } [/tex]
b=[tex] \sqrt{9+2 \sqrt{14} } [/tex]
a) aratati ca [tex] (a-b)^{2} [/tex]=8
b) Calculati [tex] (a-b+2 \sqrt{2} )^{100} [/tex]

Răspuns :

O rezolvare, ar fi aceasta:

[tex]a=\sqrt{9-2\sqrt{14}}=\sqrt{(\sqrt7-\sqrt2)^2}=|\sqrt7-\sqrt2|=\sqrt7-\sqrt2[/tex]

[tex]b=\sqrt{9+2\sqrt{14}}=\sqrt{(\sqrt7+\sqrt2)^2}=|\sqrt7+\sqrt2|=\sqrt7+\sqrt2[/tex]

[tex]a-b=\sqrt7-\sqrt2-\sqrt7-\sqrt2=-2\sqrt2\Rightarrow(a-b)^2=8[/tex]

[tex](a-b+2\sqrt2)^{100}=(-2\sqrt2+2\sqrt2)^{100}=0[/tex]

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