Question

Упростить выражения См. фото

Answer 1

1.059.

$sqrt{ frac{ sqrt{2} }{a}+ frac{a}{ sqrt{2} } +2 }= sqrt{ frac{( sqrt{2})^2 +a^2+2a sqrt{2} }{a sqrt{2} } }= sqrt{ frac{(a+ sqrt{2})^2 }{a sqrt{2} } }= frac{a+ sqrt{2} }{ sqrt{acdot sqrt[4]{2} } }$

$a sqrt{2a}- sqrt[4]{8a^4}= a sqrt{2a}-sqrt[4]{(2 sqrt{2} a^2)^2}=a sqrt{2a}-sqrt{2 sqrt{2} a^2}= \ \ =a sqrt{2}cdot( sqrt{a} - sqrt[4]{2})$

$frac{a+ sqrt{2} }{ sqrt{a}cdot sqrt[4]{2} } - frac{a^2 sqrt[4]{2} -2 sqrt{a} }{a sqrt{2}cdot( sqrt{a}- sqrt[4]{2}) } = frac{(a +sqrt{2})( sqrt{a}- sqrt[4]{2})cdot sqrt{a}cdot sqrt[4]{2} -a^2 sqrt[4]{2}+2 sqrt{a}}{a sqrt{2}cdot( sqrt{a}- sqrt[4]{2})}=$

$= frac{a^2 sqrt[4]{2}-a sqrt{a}cdot sqrt{2}+a sqrt{2}cdot sqrt[4]{2}- sqrt{2}cdot sqrt{a}cdot sqrt{2}-a^2 sqrt{2}+2 sqrt{a} }{a sqrt{2}cdot( sqrt{a}- sqrt[4]{2})}= \ \ = frac{-a sqrt{a}cdot sqrt{2}+a sqrt{2}cdot sqrt[4]{2}}{a sqrt{2}cdot( sqrt{a}- sqrt[4]{2})}= frac{ sqrt{a}cdot sqrt{2}( sqrt[4]{2}- sqrt{a})}{a sqrt{2}cdot( sqrt{a} - sqrt[4]{2}) } =-1$

1.058.

$frac{xcdot frac{1}{ sqrt{x^2-a^2} +1} }{ frac{a}{ sqrt{x-a} }+ sqrt{x-a} }= frac{x+ sqrt{x^2-a^2} }{ sqrt{x^2-a^2} } : frac{a+(x-a)}{ sqrt{x-a} } =frac{x+ sqrt{x^2-a^2} }{ sqrt{(x-a)(x+a)} } cdot frac{ sqrt{x-a} }{x}=frac{x+ sqrt{x^2-a^2} }{ xsqrt{x+a} }$


$frac{x+ sqrt{x^2-a^2} }{ xsqrt{x+a} } :frac{a^2 sqrt{x+a}} {x- sqrt{x^2-a^2} } =frac{x+ sqrt{x^2-a^2} }{ xsqrt{x+a} } cdot frac{x- sqrt{x^2-a^2} }{a^2 sqrt{x+a} }= frac{x^2-(x^2-a^2)}{a^2x(x+a)}= frac{1}{x(x+a)}$


$frac{1}{x(x+a)} + frac{1}{x^2-ax}= frac{1}{x(x+a)} + frac{1}{x(x-a)}= \ \ = frac{x-a+x+a}{x(x+a)(x-a)}= frac{2}{(x+a)(x-a)}$