Упр.957 ГДЗ Никольский Потапов 7 класс (Алгебра)

957 а) (x2/y2+y/x) : (x/y2-1/y+1/x) = x+y;
б) (x/y-y/x):(x/y+y/x-2):(1+y/x) = x/(x-y);
в) (m+1-1/(1-m):(m-m2/(m-1))=-m;
г) (a-4ab/(a+b) +b):(a/(a+b) — b/(b-a) — 2ab/(a2-b2))=a-b.

1. а)

   $$\left(\frac{x^2}{y^2}+\frac{y}{x}\right):\left(\frac{x}{y^2}-\frac{1}{y}+\frac{1}{x}\right)$$
   $$=\frac{x^3+y^3}{xy^2}:\frac{x^2-xy+y^2}{xy^2}$$
   $$=\frac{x^3+y^3}{xy^2}\cdot\frac{xy^2}{x^2-xy+y^2}$$
   $$=\frac{(x+y)(x^2-xy+y^2)}{x^2-xy+y^2}=x+y.$$

2. б)

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

3. в)

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

4. г)

   $$\left(a-\frac{4ab}{a+b}+b\right):\left(\frac{a}{a+b}-\frac{b}{b-a}-\frac{2ab}{a^2-b^2}\right)$$
   $$=\frac{a(a+b)-4ab+b(a+b)}{a+b}:\left(\frac{a}{a+b}+\frac{b}{a-b}-\frac{2ab}{(a-b)(a+b)}\right)$$
   $$=\frac{a^2-2ab+b^2}{a+b}:\frac{a^2-ab+ab+b^2-2ab}{(a-b)(a+b)}$$
   $$=\frac{(a-b)^2}{a+b}:\frac{a^2-2ab+b^2}{(a-b)(a+b)}$$
   $$=\frac{(a-b)^2}{a+b}\cdot\frac{(a-b)(a+b)}{(a-b)^2}=a-b.$$

Ответ

а) $$x+y$$; б) $$\frac{x}{x-y}$$; в) $$-m$$; г) $$a-b$$.