Question

Salut ma puteti ajuta va rog cu un exercitiu:
$\frac{1}{2 + \sqrt{3} } - \frac{1}{2 - \sqrt{3} }$
Multumesc

Answer 1

   
[tex]\displaystyle\\ \frac{1}{2+\sqrt{3}}-\frac{1}{2-\sqrt{3}}=~~\text{(Rationalizam numitorii.)}\\\\ =\frac{1\times(2-\sqrt{3})}{(2+\sqrt{3})\times(2-\sqrt{3})}-\frac{1\times(2+ \sqrt{3})}{(2-\sqrt{3})\times (2+\sqrt{3})}=\\\\ =\frac{2-\sqrt{3}}{2^2-\Big(\sqrt{3}\Big)^2 }-\frac{2+\sqrt{3}}{2^2-\Big(\sqrt{3}\Big)^2 }=\\\\ =\frac{2-\sqrt{3}}{ 4-3}-\frac{2+\sqrt{3}}{4-3}=\\\\ =\frac{2-\sqrt{3}}{1}-\frac{2+\sqrt{3}}{1}=\\\\ =2-\sqrt{3}-(2+\sqrt{3})=2-\sqrt{3}-2-\sqrt{3}=\boxed{\bf -2\sqrt{3}}[/tex]


Daca intre fractii ar fi fost semnul "+" atunci s-ar fi redus radicalii si rezultatul final ar fi fost  4 ∈ N.