Question

Calculați :
$\sqrt{11 } + | \: \sqrt{11} \: - 2 \sqrt{3} \: |$
$\sqrt{41} + \: | \: \sqrt{41} - 3 \sqrt{5} \: |$
Va rog ajutați-mă ​

Answer 1

$\it \sqrt{11}<\sqrt{12} \Rightarrow \sqrt{11}<\sqrt{4\cdot3} \Rightarrow \sqrt{11}<2\sqrt3 \Rightarrow \sqrt11-2\sqrt3<0 \Rightarrow \\ \\ \Rightarrow |\sqrt11-2\sqrt3|=-(\sqrt{11}-2\sqrt3)=2\sqrt3-\sqrt{11}\ \ \ \ \ (*)\\ \\ \\ \sqrt{11}+|\sqrt{11}-2\sqrt3|\ \stackrel{(*)}{=}\ \sqrt{11}+2\sqrt3-\sqrt{11}=2\sqrt3$

__________________________________________________

____________________________________

$\it |\sqrt{41}-3\sqrt5|=|\sqrt{41}-\sqrt{3^2\cdot5}|=|\underbrace{\it \sqrt{41}-\sqrt{45}}_{<0}|=\sqrt{45}-\sqrt{41}=\\ \\ =3\sqrt5-\sqrt{41}\ \ \ \ \ (*)\\ \\ \\ \sqrt{41}+|\sqrt{41}-3\sqrt5|=\sqrt{41}+3\sqrt5-\sqrt{41}=3\sqrt5$