Упр.177 ГДЗ Мерзляк Полонский 8 класс (Алгебра)

Выполните действия:

1. $$\frac{b+4}{b^2-6b+9}:\frac{b^2-16}{2b-6}-\frac{2}{b-4}$$;
2. $$\left(\frac{m-1}{m+1}-\frac{m+1}{m-1}\right):\frac{4m}{m^2-1}$$;

Упростите выражение:

1. $$\left(x+\frac{x}{y}\right):\left(x-\frac{x}{y}\right)$$;
2. $$\left(\frac{a}{b}+\frac{a+b}{a-b}\right)\cdot\frac{ab^2}{a^2+b^2}$$;

1. $$\frac{b+4}{b^2-6b+9}:\frac{b^2-16}{2b-6}-\frac{2}{b-4}$$

   $$b^2-6b+9=(b-3)^2,\quad b^2-16=(b-4)(b+4),\quad 2b-6=2(b-3)$$

   $$\frac{b+4}{(b-3)^2}:\frac{(b-4)(b+4)}{2(b-3)}-\frac{2}{b-4}$$

   $$\frac{b+4}{(b-3)^2}\cdot\frac{2(b-3)}{(b-4)(b+4)}-\frac{2}{b-4} =\frac{2}{(b-3)(b-4)}-\frac{2}{b-4}$$

   $$\frac{2-2(b-3)}{(b-3)(b-4)}=\frac{8-2b}{(b-3)(b-4)}=\frac{2(4-b)}{(b-3)(b-4)}=\frac{2}{3-b}$$

2. $$\left(\frac{m-1}{m+1}-\frac{m+1}{m-1}\right):\frac{4m}{m^2-1}$$

   $$\frac{(m-1)^2-(m+1)^2}{(m+1)(m-1)}:\frac{4m}{(m-1)(m+1)}$$

   $$\frac{m^2-2m+1-m^2-2m-1}{(m+1)(m-1)}:\frac{4m}{(m-1)(m+1)} =\frac{-4m}{(m+1)(m-1)}:\frac{4m}{(m-1)(m+1)}$$

   $$\frac{-4m}{(m+1)(m-1)}\cdot\frac{(m-1)(m+1)}{4m}=-1$$

3. $$\left(x+\frac{x}{y}\right):\left(x-\frac{x}{y}\right)$$

   $$\frac{x(y+1)}{y}:\frac{x(y-1)}{y} =\frac{x(y+1)}{y}\cdot\frac{y}{x(y-1)}=\frac{y+1}{y-1}$$

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

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

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

Ответ

1) $$\frac{2}{3-b}$$; 2) $$-1$$; 3) $$\frac{y+1}{y-1}$$; 4) $$\frac{ab}{a-b}$$