Упр.1.100 ГДЗ Дорофеев Суворова 8 класс (Алгебра)

а) (m+3+9/(m-3)) :(m/(m-3)+3m/(3-m)^2 );
б) (n/(1+2n+n^2 )-n/(n+1)) :(1/(n+1)+n-1);
в) (x^3/y^3 +1) :(x/y^2 -1/y+1/x);
г) (1+v/u+v^2/u^2 ) :(1/v-v^2/u^3 ).

1. а)

   $$\left(m+3+\frac{9}{m-3}\right):\left(\frac{m}{m-3}+\frac{3m}{(3-m)^2}\right)$$
   $$=\frac{(m+3)(m-3)+9}{m-3}:\frac{m(m-3)+3m}{(m-3)^2}$$
   $$=\frac{m^2-9+9}{m-3}:\frac{m^2-3m+3m}{(m-3)^2}$$
   $$=\frac{m^2}{m-3}:\frac{m^2}{(m-3)^2}$$
   $$=\frac{m^2}{m-3}\cdot\frac{(m-3)^2}{m^2}=m-3.$$

2. б)

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

3. в)

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

4. г)

   $$\left(1+\frac{v}{u}+\frac{v^2}{u^2}\right):\left(\frac{1}{v}-\frac{v^2}{u^3}\right)$$
   $$=\frac{u^2+uv+v^2}{u^2}:\frac{u^3-v^3}{u^3v}$$
   $$=\frac{u^2+uv+v^2}{u^2}:\frac{(u-v)(u^2+uv+v^2)}{u^3v}$$
   $$=\frac{u^2+uv+v^2}{u^2}\cdot\frac{u^3v}{(u-v)(u^2+uv+v^2)}$$
   $$=\frac{uv}{u-v}.$$

Ответ

а) $$m-3$$; б) $$-\frac{1}{n+1}$$; в) $$\frac{x(x+y)}{y}$$; г) $$\frac{uv}{u-v}$$.