Question

A=$\frac{ \sqrt{3}-1 }{ \sqrt{3} } + \frac{ \sqrt{4} - \sqrt{3} }{ \sqrt{12} } + \frac{ \sqrt{5}- \sqrt{4} }{ \sqrt{20} } + \frac{ \sqrt{6}- \sqrt{5} }{ \sqrt{30} }$
a) Calculati numarul A
b) Aratati ca numarul A indeplineste conditia 0<A<1

Answer 1

=[√3(√3-1)/3]+[(2-√3)/2√3]+[(√5-2)/2√5]+[√30(√6-√5)/30]
=[(3-√3)/3]+[√3(2-√3)/6]+[√5(√5-2)/10]+[(√180-√150)/30]
=[(3-√3)/3]+[(2√3-3)/6]+[(5-2√5)/10]+[(6√5-5√6)/30] aducem la acelasi numitor
=[10(3-√3)+5(2√3-3)+3(5-2√5)+6√5-5√6)]/30
=(30-10√3+10√3-15+15-6√6+6√5-5√6)/30
=(30-5√6)/30
=5(6-√6)/30
=(6-√6)/6=1-√6/6

Answer 2

[tex]A=\frac{\sqrt3-1}{\sqrt3}+\frac{\sqrt4-\sqrt3}{\sqrt{12}}+\frac{\sqrt5-\sqrt4}{\sqrt{20}}+\frac{\sqrt6-\sqrt5}{\sqrt{30}}\\ \\ A=\frac{3-\sqrt3}{3}+\frac{\sqrt3(2-\sqrt3)}{6}+\frac{\sqrt5(\sqrt5-2)}{10}+\frac{\sqrt{30}(\sqrt6-\sqrt5)}{30}\\ \\ A=\frac{3-\sqrt3}{3}+\frac{2\sqrt3-3}{6}+\frac{5-2\sqrt5}{10}+\frac{6\sqrt5-5\sqrt6}{30}\\ \\ A=\frac{30-10\sqrt3+10\sqrt3-15+15-6\sqrt5+6\sqrt5-5\sqrt6}{30}\\ A=\frac{30-5\sqrt6}{30}\\ A=\frac{6-\sqrt6}{6}\\ b)0\ \textless \ A\ \textless \ 1\\ 0\ \textless \ \frac{6-\sqrt6}{6}\ \textless \ 1\\ 0 \ \textless \ 6-\sqrt6\ \textless \ 6[/tex]
[tex]-6\ \textless \ -\sqrt6\ \textless \ 0\\ 6\ \textgreater \ \sqrt6\ \textgreater \ 0|()^2\\ 36\ \textgreater \ 6\ \textgreater \ 0 (A)[/tex]