Question

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

Answer 1

O rezolvare, ar fi aceasta:

$a=\sqrt{9-2\sqrt{14}}=\sqrt{(\sqrt7-\sqrt2)^2}=|\sqrt7-\sqrt2|=\sqrt7-\sqrt2$

$b=\sqrt{9+2\sqrt{14}}=\sqrt{(\sqrt7+\sqrt2)^2}=|\sqrt7+\sqrt2|=\sqrt7+\sqrt2$

$a-b=\sqrt7-\sqrt2-\sqrt7-\sqrt2=-2\sqrt2\Rightarrow(a-b)^2=8$

$(a-b+2\sqrt2)^{100}=(-2\sqrt2+2\sqrt2)^{100}=0$