Question

Vreau Exercițiile 2,,4,5 dacă se poate

Answer 1

$\displaystyle\bf\\\boxed{\bf \sqrt[2k]{x}~exista~daca~si~numai~daca~x\in\mathbb{R}_{+},~2k\in\mathbb{N} }~.\\\\\ 3.~\sqrt{\bigg(a-\sqrt{20}\bigg)^2}+\sqrt{\bigg(b+\sqrt{80}\bigg)^2}+\sqrt{\bigg(c-\sqrt{180}\bigg)^2} \leq 0.~a,~b,~c~\in\mathbb{R}.\\cum~cei~trei~termeni~ai~sumei~sunt~mai~mari~sau~egali~cu~0,~dar~suma~\\trebuie~sa~fie~mai~mica~sau~egala~cu~0\implies \boxed{\bf termenii~sumei~sunt~nuli}.\\\\\sqrt{\bigg(a-\sqrt{20}\bigg)^2} = 0 \Leftrightarrow a-\sqrt{20}=0 \implies a=\sqrt{20}=2\sqrt{5}.$

$\displaystyle\bf\\\sqrt{\bigg(b+\sqrt{80}\bigg)^2}=0 \Leftrightarrow b+\sqrt{80}=0 \implies b=-\sqrt{80}=-4\sqrt{5}.\\\sqrt{\bigg(c-\sqrt{180}\bigg)^2}=0 \Leftrightarrow c-\sqrt{180}=0 \implies c=\sqrt{180}=6\sqrt{5}.\\a+b+c=2\sqrt{5}-4\sqrt{5}+6\sqrt{5}=\boxed{\bf 4\sqrt{5}}~.$