Упр.1.94 ГДЗ Дорофеев Суворова 8 класс (Алгебра)

а) (1-u/(u+v))•(1+v/(u-v));
б) (1/(x+z)-1/(x-z)) :(1/(x+z)+1/(x-z));
в) (a/b+b/a+2) :(a/b-b/a);
г) (1/y-y)•(1/(y+1)-1/(y-1)).

1. а)

   $$\left(1-\frac{u}{u+v}\right)\cdot\left(1+\frac{v}{u-v}\right)=\frac{u+v-u}{u+v}\cdot\frac{u-v+v}{u-v}=\frac{v}{u+v}\cdot\frac{u}{u-v}=\frac{uv}{u^2-v^2}.$$

2. б)

   $$\left(\frac{1}{x+z}-\frac{1}{x-z}\right):\left(\frac{1}{x+z}+\frac{1}{x-z}\right)=\frac{x-z-(x+z)}{(x+z)(x-z)}:\frac{x-z+(x+z)}{(x+z)(x-z)}$$
   $$=\frac{-2z}{x^2-z^2}:\frac{2x}{x^2-z^2}=\frac{-2z}{x^2-z^2}\cdot\frac{x^2-z^2}{2x}=-\frac{z}{x}.$$

3. в)

   $$\left(\frac{a}{b}+\frac{b}{a}+2\right):\left(\frac{a}{b}-\frac{b}{a}\right)=\frac{a^2+b^2+2ab}{ab}:\frac{a^2-b^2}{ab}$$
   $$=\frac{(a+b)^2}{ab}\cdot\frac{ab}{a^2-b^2}=\frac{(a+b)^2}{(a-b)(a+b)}=\frac{a+b}{a-b}.$$

4. г)

   $$\left(\frac{1}{y}-y\right)\cdot\left(\frac{1}{y+1}-\frac{1}{y-1}\right)=\frac{1-y^2}{y}\cdot\frac{y-1-(y+1)}{(y+1)(y-1)}$$
   $$=\frac{1-y^2}{y}\cdot\frac{-2}{y^2-1}=\frac{1-y^2}{y}\cdot\frac{2}{1-y^2}=\frac{2}{y}.$$

Ответ

а) $$\frac{uv}{u^2-v^2}$$; б) $$-\frac{z}{x}$$; в) $$\frac{a+b}{a-b}$$; г) $$\frac{2}{y}$$.