Răspuns:
1. [tex][\sqrt{\frac{(\sqrt{7} - \sqrt{5})(\sqrt{7} - \sqrt{5}) }{(\sqrt{7} + \sqrt{5})(\sqrt{7} - \sqrt{5})}}+ \sqrt{\frac{(\sqrt{5} - \sqrt{3})(\sqrt{5} - \sqrt{3}) }{(\sqrt{5} + \sqrt{3})(\sqrt{5} - \sqrt{3})}} + \sqrt{\frac{(\sqrt{3} - 1)(\sqrt{3} -1) }{(\sqrt{3} + 1)(\sqrt{3} - 1)}}*\frac{\sqrt{2} (\sqrt{7}+1)}{2\sqrt{2}} =[/tex]
2. [tex]=(\sqrt \frac{7-2\sqrt{35} +5}{7-5} +\sqrt \frac{5-2\sqrt{15} +3}{5-2} \sqrt \frac{3-2\sqrt{3} +}{3-1} )*\frac{\sqrt{7} +1}{2} =\\[/tex]
3. [tex]=[\sqrt{\frac{2(6-\sqrt{35})}{2}} +\sqrt{\frac{2(4-\sqrt{15})}{2}} +\sqrt{\frac{2(2-\sqrt{3})}{2}}*[tex]=(\sqrt{\frac{5}{2}} -\sqrt{\frac{7}{2}}+\sqrt{\frac{3}{2}}-\sqrt{\frac{5}{2}}+\sqrt{\frac{1}{2}}-\sqrt{\frac{3}{2}})*\frac{\sqrt{7} +1}{2} =[/tex][/tex]
4.[tex]= (\sqrt{6-\sqrt{35}} +\sqrt{4-\sqrt{15}} +\sqrt{2-\sqrt{3}}} *\frac{\sqrt{7} +1}{2}-[/tex]
5. [tex]=(\sqrt{\frac{6+\sqrt{36-35}}{2}} - \sqrt{\frac{6-\sqrt{36-35}}{2}} + \sqrt{\frac{4+\sqrt{16-15}}{2}} - \sqrt{\frac{4-\sqrt{16-15}}{2}} \sqrt{\frac{2+\sqrt{4-3}}{2}} - \sqrt{\frac{2-\sqrt{4-3}}{2}} )*\frac{\sqrt{7}+1}{2} =[/tex]
6. [tex]=(\sqrt{\frac{5}{2}}-\sqrt{\frac{7}{2}} +\sqrt{\frac{3}{2}}-\sqrt{\frac{5}{2}} +\sqrt{\frac{1}{2}}-\sqrt{\frac{3}{2}})*\frac{\sqrt{7}+1}{2}=[/tex]
[tex]=\frac{1-\sqrt{7} }{\sqrt{2}} *\frac{\sqrt{7}+1}{2} =[/tex]
7. [tex]=\frac{\sqrt{2} (1-7)}{2}=-\frac{6\sqrt{2}}{2} =-3\sqrt{2}[/tex]
Explicație pas cu pas:
1. La deimpartit se rationalizeaza numitorii folosind formula de calcul prescurtat (a+b)(a-b)=a²-b², se rastoarna impartitorul si se da factor comun [tex]\sqrt{2}[/tex] la numarator.
2.La deimpartit se aplica formula de calcul prescurtat (a-b)²=a²-2ab+b², iar la impartitor se reduce [tex]\sqrt{2}[/tex]
3. Se efectueaza adunarile si se da 2 factor comun la fiecare numarator.
4. Se reduce 2 sub fiecare radical
5. Se aplica formula de calcul prescurtat [tex]\sqrt{a-\sqrt{b}} =\sqrt{\frac{a+\sqrt{a^{2}-b}}{2}} - \sqrt{\frac{a-\sqrt{a^{2}-b}}{2}}[/tex]
6. Se efectueaza calculele de la numaratori si apoi cele din paranteza
7. Se rationalizeaza primul numitor si se aplica formula (a+b)(a-b)=a²-b². E efectueaza calculul din paranteza si se simplifica cu 2.
