Упр.111 ГДЗ Мерзляк Полонский 8 класс (Алгебра)

1) (m+n)/(m-n) — (m2+n2)/(m2-n2);
2) (x-y)/(x+y) + y2/(2xy+x2+y2); Выполните действия:
1) a — 4/a;
2) 1/x + x — 2;
3) m/n2 — 1/n + m;

1. $$\frac{m+n}{m-n}-\frac{m^2+n^2}{m^2-n^2}=\frac{(m+n)^2-(m^2+n^2)}{(m-n)(m+n)}$$
   $$=\frac{m^2+2mn+n^2-m^2-n^2}{m^2-n^2}=\frac{2mn}{m^2-n^2}.$$

2. $$\frac{x-y}{x+y}+\frac{y^2}{2xy+x^2+y^2}=\frac{x-y}{x+y}+\frac{y^2}{(x+y)^2}$$
   $$=\frac{(x-y)(x+y)+y^2}{(x+y)^2}=\frac{x^2-y^2+y^2}{(x+y)^2}=\frac{x^2}{(x+y)^2}.$$

3. $$\frac{2a}{4a^2-1}-\frac{a+4}{2a^2+a}=\frac{2a}{(2a-1)(2a+1)}-\frac{a+4}{a(2a+1)}$$
   $$=\frac{2a^2-(a+4)(2a-1)}{a(2a-1)(2a+1)}$$
   $$=\frac{2a^2-2a^2+a-8a+4}{a(2a-1)(2a+1)}=\frac{4-7a}{a(4a^2-1)}.$$

4. $$\frac{b-2}{b^2+6b+9}-\frac{b}{b^2-9}=\frac{b-2}{(b+3)^2}-\frac{b}{(b-3)(b+3)}$$
   $$=\frac{(b-2)(b-3)-b(b+3)}{(b-3)(b+3)^2}$$
   $$=\frac{b^2-5b+6-b^2-3b}{(b-3)(b+3)^2}=\frac{6-8b}{(b-3)(b+3)^2}.$$

5. $$\frac{x-6}{x^2+3x}+\frac{x}{x+3}-\frac{x-3}{x}=\frac{x-6}{x(x+3)}+\frac{x}{x+3}-\frac{x-3}{x}$$
   $$=\frac{x-6+x^2-(x-3)(x+3)}{x(x+3)}$$
   $$=\frac{x-6+x^2-(x^2-9)}{x(x+3)}=\frac{x+3}{x(x+3)}=\frac{1}{x}.$$

6. $$\frac{y+2}{y-2}-\frac{y-2}{y+2}-\frac{16}{y^2-4}=\frac{(y+2)^2-(y-2)^2-16}{(y-2)(y+2)}$$
   $$=\frac{y^2+4y+4-y^2+4y-4-16}{y^2-4}=\frac{8y-16}{y^2-4}$$
   $$=\frac{8(y-2)}{(y-2)(y+2)}=\frac{8}{y+2}.$$

Ответ

1. $$\frac{2mn}{m^2-n^2}$$
2. $$\frac{x^2}{(x+y)^2}$$
3. $$\frac{4-7a}{a(4a^2-1)}$$
4. $$\frac{6-8b}{(b-3)(b+3)^2}$$
5. $$\frac{1}{x}$$
6. $$\frac{8}{y+2}$$