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![frac{2+ sqrt{3} }{ sqrt{2}+ sqrt{2+ sqrt{3} } }+ frac{2- sqrt{3} }{ sqrt{2}- sqrt{2- sqrt{3} } }= \ \ = frac{(2+ sqrt{3})cdot ( sqrt{2}- sqrt{2+ sqrt{3} } ) }{ (sqrt{2}+ sqrt{2+ sqrt{3} })( sqrt{2}- sqrt{2+ sqrt{3} } ) }+ frac{(2- sqrt{3})( sqrt{2}+ sqrt{2- sqrt{3} } }{( sqrt{2}- sqrt{2- sqrt{3} )( sqrt{2}+ sqrt{2- sqrt{3} } }) }= \ \ = frac{2+ sqrt{3} }{ sqrt{2}+ sqrt{2+ sqrt{3} } }+ frac{2- sqrt{3} }{ sqrt{2}- sqrt{2- sqrt{3} } }= \ \ = frac{(2+ sqrt{3})cdot ( sqrt{2}- sqrt{2+ sqrt{3} } ) }{ (sqrt{2}+ sqrt{2+ sqrt{3} })( sqrt{2}- sqrt{2+ sqrt{3} } ) }+ frac{(2- sqrt{3})( sqrt{2}+ sqrt{2- sqrt{3} } }{( sqrt{2}- sqrt{2- sqrt{3} )( sqrt{2}+ sqrt{2- sqrt{3} } }) }= \ \ =](https://tex.z-dn.net/?f=+frac%7B2%2B+sqrt%7B3%7D+%7D%7B+sqrt%7B2%7D%2B+sqrt%7B2%2B+sqrt%7B3%7D+%7D++%7D%2B+frac%7B2-+sqrt%7B3%7D+%7D%7B+sqrt%7B2%7D-+sqrt%7B2-+sqrt%7B3%7D+%7D++%7D%3D+%5C++%5C+%3D++frac%7B%282%2B+sqrt%7B3%7D%29cdot+%28+sqrt%7B2%7D-+sqrt%7B2%2B+sqrt%7B3%7D+%7D+%29+%7D%7B+%28sqrt%7B2%7D%2B+sqrt%7B2%2B+sqrt%7B3%7D+%7D%29%28+sqrt%7B2%7D-+sqrt%7B2%2B+sqrt%7B3%7D+%7D+%29++%7D%2B+frac%7B%282-+sqrt%7B3%7D%29%28+sqrt%7B2%7D%2B+sqrt%7B2-+sqrt%7B3%7D+%7D++%7D%7B%28+sqrt%7B2%7D-+sqrt%7B2-+sqrt%7B3%7D+%29%28+sqrt%7B2%7D%2B+sqrt%7B2-+sqrt%7B3%7D+%7D+%7D%29++%7D%3D+%5C++%5C+%3D)
![frac{(2+ sqrt{3})cdot ( sqrt{2}- sqrt{2+ sqrt{3} } ) }{ (sqrt{2})^2-( sqrt{2+ sqrt{3} })^2 }+ frac{(2- sqrt{3})( sqrt{2}+ sqrt{2- sqrt{3} } }{( sqrt{2})^2-( sqrt{2- sqrt{3} })^2 }= \ \ = frac{(2+ sqrt{3})cdot ( sqrt{2}- sqrt{2+ sqrt{3} } ) }{ 2-(2+ sqrt{3})}+frac{(2- sqrt{3})( sqrt{2}+ sqrt{2- sqrt{3} } }{2-( 2- sqrt{3} ) }= frac{(2+ sqrt{3})cdot ( sqrt{2}- sqrt{2+ sqrt{3} } ) }{ (sqrt{2})^2-( sqrt{2+ sqrt{3} })^2 }+ frac{(2- sqrt{3})( sqrt{2}+ sqrt{2- sqrt{3} } }{( sqrt{2})^2-( sqrt{2- sqrt{3} })^2 }= \ \ = frac{(2+ sqrt{3})cdot ( sqrt{2}- sqrt{2+ sqrt{3} } ) }{ 2-(2+ sqrt{3})}+frac{(2- sqrt{3})( sqrt{2}+ sqrt{2- sqrt{3} } }{2-( 2- sqrt{3} ) }=](https://tex.z-dn.net/?f=+frac%7B%282%2B+sqrt%7B3%7D%29cdot+%28+sqrt%7B2%7D-+sqrt%7B2%2B+sqrt%7B3%7D+%7D+%29+%7D%7B+%28sqrt%7B2%7D%29%5E2-%28+sqrt%7B2%2B+sqrt%7B3%7D+%7D%29%5E2++%7D%2B+frac%7B%282-+sqrt%7B3%7D%29%28+sqrt%7B2%7D%2B+sqrt%7B2-+sqrt%7B3%7D+%7D++%7D%7B%28+sqrt%7B2%7D%29%5E2-%28+sqrt%7B2-+sqrt%7B3%7D+%7D%29%5E2++%7D%3D+%5C++%5C+%3D+frac%7B%282%2B+sqrt%7B3%7D%29cdot+%28+sqrt%7B2%7D-+sqrt%7B2%2B+sqrt%7B3%7D+%7D+%29+%7D%7B+2-%282%2B+sqrt%7B3%7D%29%7D%2Bfrac%7B%282-+sqrt%7B3%7D%29%28+sqrt%7B2%7D%2B+sqrt%7B2-+sqrt%7B3%7D+%7D++%7D%7B2-%28+2-+sqrt%7B3%7D+%29++%7D%3D)
![=frac{(2+ sqrt{3})cdot ( sqrt{2}- sqrt{2+ sqrt{3} } ) }{- sqrt{3}}+frac{(2- sqrt{3})( sqrt{2}+ sqrt{2- sqrt{3} } )}{ sqrt{3} }= \ \ =frac{(2+ sqrt{3})cdot ( sqrt{2}- sqrt{2+ sqrt{3} } )cdot (- sqrt{3}) }{3}+frac{(2- sqrt{3})( sqrt{2}+ sqrt{2- sqrt{3} })cdot sqrt{3} }{ 3 }= =frac{(2+ sqrt{3})cdot ( sqrt{2}- sqrt{2+ sqrt{3} } ) }{- sqrt{3}}+frac{(2- sqrt{3})( sqrt{2}+ sqrt{2- sqrt{3} } )}{ sqrt{3} }= \ \ =frac{(2+ sqrt{3})cdot ( sqrt{2}- sqrt{2+ sqrt{3} } )cdot (- sqrt{3}) }{3}+frac{(2- sqrt{3})( sqrt{2}+ sqrt{2- sqrt{3} })cdot sqrt{3} }{ 3 }=](https://tex.z-dn.net/?f=%3Dfrac%7B%282%2B+sqrt%7B3%7D%29cdot+%28+sqrt%7B2%7D-+sqrt%7B2%2B+sqrt%7B3%7D+%7D+%29+%7D%7B-+sqrt%7B3%7D%7D%2Bfrac%7B%282-+sqrt%7B3%7D%29%28+sqrt%7B2%7D%2B+sqrt%7B2-+sqrt%7B3%7D+%7D++%29%7D%7B+sqrt%7B3%7D+++%7D%3D+%5C++%5C+%3Dfrac%7B%282%2B+sqrt%7B3%7D%29cdot+%28+sqrt%7B2%7D-+sqrt%7B2%2B+sqrt%7B3%7D+%7D+%29cdot+%28-+sqrt%7B3%7D%29++%7D%7B3%7D%2Bfrac%7B%282-+sqrt%7B3%7D%29%28+sqrt%7B2%7D%2B+sqrt%7B2-+sqrt%7B3%7D+%7D%29cdot++sqrt%7B3%7D+%7D%7B+3+++%7D%3D)
![= frac{1}{3}cdot (-2 sqrt{6}-3 sqrt{2} +2 sqrt{3}cdot sqrt{2+ sqrt{3} } +3 sqrt{2+ sqrt{3} }+2 sqrt{6}-3 sqrt{2}+ \ \ +2 sqrt{3}cdot sqrt{2- sqrt{3} }-3 sqrt{2- sqrt{3} })= = frac{1}{3}cdot (-2 sqrt{6}-3 sqrt{2} +2 sqrt{3}cdot sqrt{2+ sqrt{3} } +3 sqrt{2+ sqrt{3} }+2 sqrt{6}-3 sqrt{2}+ \ \ +2 sqrt{3}cdot sqrt{2- sqrt{3} }-3 sqrt{2- sqrt{3} })=](https://tex.z-dn.net/?f=%3D+frac%7B1%7D%7B3%7Dcdot+%28-2+sqrt%7B6%7D-3+sqrt%7B2%7D++%2B2+sqrt%7B3%7Dcdot+sqrt%7B2%2B+sqrt%7B3%7D+%7D+%2B3+sqrt%7B2%2B+sqrt%7B3%7D+%7D%2B2+sqrt%7B6%7D-3+sqrt%7B2%7D%2B+%5C++%5C+%2B2+sqrt%7B3%7Dcdot+sqrt%7B2-+sqrt%7B3%7D+%7D-3+sqrt%7B2-+sqrt%7B3%7D+%7D%29%3D++++++++)
![= frac{1}{3}cdot (-6sqrt{2} +2 sqrt{2sqrt{3}+3 } +3 sqrt{2+ sqrt{3} } +2cdot sqrt{2sqrt{3}-3 }-3 sqrt{2- sqrt{3} }) = frac{1}{3}cdot (-6sqrt{2} +2 sqrt{2sqrt{3}+3 } +3 sqrt{2+ sqrt{3} } +2cdot sqrt{2sqrt{3}-3 }-3 sqrt{2- sqrt{3} })](https://tex.z-dn.net/?f=%3D+frac%7B1%7D%7B3%7Dcdot+%28-6sqrt%7B2%7D++%2B2+sqrt%7B2sqrt%7B3%7D%2B3+%7D+%2B3+sqrt%7B2%2B+sqrt%7B3%7D+%7D+%2B2cdot+sqrt%7B2sqrt%7B3%7D-3+%7D-3+sqrt%7B2-+sqrt%7B3%7D+%7D%29++++++++)
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