Question

Пожалуйста помогите решить примеры из номера 566.

Answer 1

№566
1)
$frac{ax}{a+x} - frac{bx}{b-x}= frac{ax(b-x)-bx(a+x)}{(a+x)(b-x)}= frac{axb-ax^2-bxa-bx^2}{(a+x)(b-x)}= frac{-x^2(a+b)}{(a+x)(b-x)}$
$x= frac{ab}{a-b}$
$frac{-x^2(a+b)}{(a+x)(b-x)} = frac{-(frac{ab}{a-b})^2(a+b)}{(a+frac{ab}{a-b})(b-frac{ab}{a-b})} =frac{-(frac{ab}{a-b})^2(a+b)}{frac{a^2-ab+ab}{a-b}*frac{ab-b^2-ab}{a-b}} =frac{-(frac{ab}{a-b})^2(a+b)}{frac{a^2}{a-b}*(frac{-b^2}{a-b})} =$$=frac{-frac{(ab)^2}{(a-b)^2}*(a+b)}{-frac{a^2b^2}{(a-b)^2}} = frac{a^2b^2*(a+b)}{(a-b)^2}* frac{(a-b)^2}{a^2b^2}= a+b$
2)
$frac{x^2y^2}{ x^{2} -y^2}$   при  $x= frac{2ab}{a^2-b^2}$  $y= frac{2ab}{a^2+b^2}$
$frac{ (frac{2ab}{a^2-b^2} )^2*( frac{2ab}{a^2+b^2})^2 }{( frac{2ab}{a^2-b^2})^2-(frac{2ab}{a^2+b^2})^2 }= frac{ (frac{2ab}{a^2-b^2} * frac{2ab}{a^2+b^2})^2 }{( frac{2ab}{a^2-b^2}-frac{2ab}{a^2+b^2})(frac{2ab}{a^2-b^2}+frac{2ab}{a^2+b^2} ) }=$$= frac{ (frac{4a^2b^2}{(a^2-b^2)(a^2+b^2)} )^2 }{frac{2ab(a^2+b^2-a^2+b^2)}{(a^2-b^2)(a^2+b^2)}*frac{2ab(a^2+b^2+a^2-b^2)}{(a^2-b^2)(a^2+b^2)}}= frac{ (frac{4a^2b^2}{a^4-b^4} )^2 }{frac{2ab*2b^2}{a^4-b^4}*frac{2ab*2a^2}{a^4-b^4}}=$$= frac{ frac{16a^4b^4}{(a^4-b^4)^2} }{frac{4ab^3}{a^4-b^4}*frac{4a^3b}{a^4-b^4}}= frac{16a^4b^4}{(a^4-b^4)^2} * frac{(a^4-b^4)^2}{16a^4b^4} =1$