So sánh các số: a) √7 - √2 và 1 b) √8 - √5 và √7 + √6

a,

$(\sqrt{7}-\sqrt{2})^2= 7-2\sqrt{14}+2= 9-2\sqrt{14}$

$\Rightarrow (\sqrt7-\sqrt2)^2-1= 8-2\sqrt{14}$

Ta có $8^2=64; (2\sqrt{14})^2=56$

$\Rightarrow 8^2> (2\sqrt{14})^2$

$\Leftrightarrow 8>2\sqrt{14}$

$\Leftrightarrow 8-2\sqrt{14}>0$

$\Leftrightarrow (\sqrt7-\sqrt2)^2-1>0$

$\Leftrightarrow (\sqrt7-\sqrt2)^2>1$

$\Leftrightarrow \sqrt7-\sqrt2 > 1$ ($\sqrt7-\sqrt2>0\Rightarrow |\sqrt7-\sqrt2|=\sqrt7-\sqrt2$)

b,

$(\sqrt8-\sqrt5)^2= 8+5-2\sqrt{8.5}=13-2\sqrt{40}$

$(\sqrt7+\sqrt6)^2=7+6+2\sqrt{7.6}=13+2\sqrt{42}$

$-2\sqrt{40} < 2\sqrt{42}$

$\Leftrightarrow 13-2\sqrt{40}<13+2\sqrt{42}$

$\Leftrightarrow (\sqrt8-\sqrt5)^2<(\sqrt7+\sqrt6)^2$

$\Leftrightarrow |\sqrt8-\sqrt5|<|\sqrt7+\sqrt6|$

$\Leftrightarrow \sqrt8-\sqrt5 < \sqrt7+\sqrt6$