№ 17.35 Алгебра = № 33.35 Математика
Спростіть вираз:
1. $\sqrt{(\sqrt3-5)^2}+(\sqrt{\sqrt3-1})^2;$
2. $\sqrt{(3-\sqrt7)^2}+\sqrt{(2-\sqrt7)^2};$
3. $\sqrt{(\sqrt{21}-5)^2}-(\sqrt{\sqrt{21}-4})^2;$
4. $\sqrt{7+4\sqrt3}.$
Розв'язок:
1. $\sqrt{(\sqrt3-5)^2}+(\sqrt{\sqrt3-1})^2=$
$= |\sqrt3-5|+(\sqrt3-1)=$
$= -(\sqrt3-5)+(\sqrt3-1)=$
$ =-\sqrt3+5+\sqrt3-1=4;$
2. $\sqrt{(3-\sqrt7)^2}+\sqrt{(2-\sqrt7)^2}=$
$= |3-\sqrt7|-|2-\sqrt7|=$
$= (3-\sqrt7)-(2-\sqrt7)=$
$ =3-\sqrt7+\sqrt7-2=1;$
3. $\sqrt{(\sqrt{21}-5)^2}-\sqrt{(\sqrt{21}-4)^2}=$
$= |\sqrt{21}-5|- |\sqrt{21}-4|=$
$= -(\sqrt{21}-5)-(\sqrt{21}-4)=$
$= -\sqrt{21}+5-\sqrt{21}+4=$
$= 9-2\sqrt{21};$
4. $\sqrt{7+4\sqrt3}=$
$= \sqrt{4+4\sqrt3+3}=$
$= \sqrt{2^2+2\cdot2\sqrt3+(\sqrt3)^2}=$
$= \sqrt{(2+\sqrt3)^2}=|2+\sqrt3|=$
$= 2+\sqrt3.$