1. [latex]left(frac{1}{ sqrt{5} - sqrt{6} } + frac{1}{ sqrt{5} - 2 } ight) : left(1 + frac{ sqrt{6} }{2} ight) =[/latex] [latex]left(frac{ sqrt{5} + sqrt{6} }{( sqrt{5} - sqrt{6} )( sqrt{5} + sqrt{6} )} + frac{ sqrt{5}+2 }{ (sqrt{5} - 2 )( sqrt{5}+2)} ight) : left( frac{2+ sqrt{6} }{2} ight) =[/latex] [latex]left(frac{ sqrt{5} + sqrt{6} }{5-6} + frac{ sqrt{5}+2 }{5-4} ight) cdot frac{2}{2+ sqrt{6}} =[/latex] [latex]left(frac{ sqrt{5} + sqrt{6} }{-1} + frac{ sqrt{5}+2 }{1} ight) cdot frac{2}{2+ sqrt{6}} =[/latex] [latex]left(-sqrt{5} - sqrt{6} +sqrt{5}+2 ight) cdot frac{2}{2+ sqrt{6}} =[/latex] [latex]left(2-sqrt{6} ight) cdot frac{2}{2+ sqrt{6}} =frac{2(2-sqrt{6})}{2+ sqrt{6}} =[/latex] [latex]frac{2(2-sqrt{6})^2}{(2+ sqrt{6})(2-sqrt{6})} =frac{2(2-sqrt{6})^2}{4-6} =[/latex] [latex]frac{2(2-sqrt{6})^2}{-2} =-(4-4sqrt{6}+6)=[/latex] [latex]-(10-4sqrt{6}) =-10+4sqrt{6}[/latex] ===================== 2. [latex]frac{1}{2 (1+ sqrt{2}) } + frac{1}{2 (1- sqrt{2}) }=[/latex] [latex]frac{1- sqrt{2} }{2 (1+ sqrt{2})(1- sqrt{2}) } + frac{1+ sqrt{2} }{2 (1- sqrt{2})(1+ sqrt{2}) }=[/latex] [latex]frac{1- sqrt{2} }{2 cdot (1-2) } + frac{1+ sqrt{2} }{2 cdot (1-2) }=frac{1- sqrt{2} }{2 cdot (-1) } + frac{1+ sqrt{2} }{2 cdot (-1) }=[/latex] [latex]frac{1- sqrt{2} }{-2} + frac{1+ sqrt{2} }{-2}=frac{1- sqrt{2} +1+ sqrt{2} }{-2}= frac{2}{-2}=-1[/latex]
