Question

Aratati ca numarul x este un numar intreg, unde:
$x= \sqrt{3}*( \frac{ \sqrt{6} }{3 \sqrt{2} }+ \frac{25}{2 \sqrt{75} })-( \frac{12}{5 \sqrt{3} }- \frac{ \sqrt{3} }{2}): \frac{ \sqrt{27} }{27}-( \frac{ \sqrt{48} }{5}+2 \sqrt{3})* \frac{1}{ \sqrt{3} }$
Va rog frumos sa ma ajutati pentru ca nu prea le am cu radicalii....

Answer 1

   

$\displaystyle \\ x= \sqrt{3}*( \frac{ \sqrt{6} }{3 \sqrt{2} }+ \frac{25}{2 \sqrt{75} })-( \frac{12}{5 \sqrt{3} }- \frac{ \sqrt{3} }{2}): \frac{3\sqrt{3} }{27}-( \frac{ \sqrt{48} }{5}+2 \sqrt{3})* \frac{1}{ \sqrt{3} } = \\ \\ =\sqrt{3}*( \frac{ \sqrt{6} }{3 \sqrt{2} }+ \frac{25}{2\cdot 5 \sqrt{3} })-( \frac{12}{5 \sqrt{3} }- \frac{ \sqrt{3} }{2}): \frac{3\sqrt{3} }{27}-( \frac{ 4\sqrt{3} }{5}+2 \sqrt{3})* \frac{1}{ \sqrt{3} } =$


[tex] \displaystyle \\ =\sqrt{3}*( \frac{ \sqrt{3} }{3 }+ \frac{5}{2\sqrt{3} })-( \frac{12}{5 \sqrt{3} }- \frac{ \sqrt{3} }{2}): \frac{\sqrt{3} }{9}- \sqrt{3}( \frac{ 4}{5}+2)* \frac{1}{ \sqrt{3} } = \\ \\ =\sqrt{3}*( \frac{ 2\sqrt{3} *\sqrt{3} +3*5}{3*2\sqrt{3} } )-( \frac{24-5\sqrt{3}*\sqrt{3} }{ 2*5\sqrt{3} }): \frac{\sqrt{3} }{9}- \sqrt{3}* \frac{ 14}{5}* \frac{1}{ \sqrt{3} } = [/tex]


[tex]\displaystyle \\ =\sqrt{3}*\frac{ 6 +15}{6\sqrt{3} } - \frac{24-15 }{ 10\sqrt{3} }*\frac{9 }{\sqrt{3}}- \frac{ 14\sqrt{3}}{5\sqrt{3}} = \\ \\ =\frac{ 21\sqrt{3}}{6\sqrt{3} } - \frac{9 }{ 10\sqrt{3} }*\frac{9 }{\sqrt{3}}- \frac{ 14}{5} = \\ \\ =\frac{ 7}{2 } - \frac{81 }{ 30 }- \frac{ 14}{5} = \\ \\ =\frac{ 7*15}{30 } - \frac{81 }{ 30 }- \frac{ 14*6}{30} =\frac{105 -81 - 84}{30 }= \frac{-60}{30 }= \boxed{-2} [/tex]