Упр.403 ГДЗ Никольский Потапов 9 класс (Алгебра)

а)(a^(1/2)+b^(1/2) )^(-2)+(a^(1/2)-b^(1/2) )^(-2)
б) (x^(-1/2) y^(-1/2)+x^(-1/6) (y^(-5/6)-x^(-1/3) y^(-1/2) ) )^(-3/2)
в)((a+1)/(a^(2/3)-a^(1/3)+1)-(a-1)/(a-a^(1/2) ))^2

1. а)

   $$\left(a^{-\frac12}+b^{-\frac12}\right)^{-2}+\left(a^{-\frac12}-b^{-\frac12}\right)^{-2}$$
   $$=\frac{1}{\left(a^{-\frac12}+b^{-\frac12}\right)^2}+\frac{1}{\left(a^{-\frac12}-b^{-\frac12}\right)^2}$$
   $$=\frac{\left(a^{-\frac12}-b^{-\frac12}\right)^2+\left(a^{-\frac12}+b^{-\frac12}\right)^2}{\left(a^{-\frac12}+b^{-\frac12}\right)^2\left(a^{-\frac12}-b^{-\frac12}\right)^2}$$
   $$=\frac{2a^{-1}+2b^{-1}}{\left(a^{-1}-b^{-1}\right)^2}$$
   $$=\frac{2\left(a^{-1}+b^{-1}\right)}{\left(a^{-1}-b^{-1}\right)^2}.$$

2. б)

   $$\left(x^{-\frac12}y^{-\frac12}+x^{-\frac16}\left(y^{-\frac56}-x^{-\frac13}y^{-\frac12}\right)\right)^{-\frac32}$$
   $$=\left(x^{-\frac12}y^{-\frac12}+x^{-\frac16}y^{-\frac56}-x^{-\frac12}y^{-\frac12}\right)^{-\frac32}$$
   $$=\left(x^{-\frac16}y^{-\frac56}\right)^{-\frac32}$$
   $$=x^{\frac14}y^{\frac54}=y\sqrt[4]{xy}.$$

3. в)

   $$\left(\frac{a+1}{a^{\frac23}-a^{\frac13}+1}-\frac{a-1}{a-a^{\frac12}}\right)^2$$
   $$=\left(\frac{(a^{\frac13}+1)(a^{\frac23}-a^{\frac13}+1)}{a^{\frac23}-a^{\frac13}+1}-\frac{(a^{\frac12}-1)(a^{\frac12}+1)}{a^{\frac12}(a^{\frac12}-1)}\right)^2$$
   $$=\left(a^{\frac13}+1-\frac{a^{\frac12}+1}{a^{\frac12}}\right)^2$$
   $$=\left(a^{\frac13}-\frac{1}{a^{\frac12}}\right)^2$$
   $$=\left(\frac{a^{\frac56}-1}{a^{\frac12}}\right)^2 =\frac{\left(a^{\frac56}-1\right)^2}{a}.$$

Ответ

а) $$\frac{2\left(a^{-1}+b^{-1}\right)}{\left(a^{-1}-b^{-1}\right)^2}$$; б) $$y\sqrt[4]{xy}$$; в) $$\frac{\left(a^{\frac56}-1\right)^2}{a}$$.