Задание 3 Параграф 4 ГДЗ Рабочая тетрадь 1 Мерзляк Полонский 8 класс (Алгебра)

Выполните сложение или вычитание дробей:

1. $$\frac{4a}{7a+49}+\frac{4}{a+7}$$;
2. $$\frac{a^2-4ab}{ab-2b^2}+\frac{4b}{a-2b}$$;
3. $$\frac{6}{2m-5}-\frac{4m+5}{2m^2-5m}$$;
4. $$\frac{1}{c^2-6c}-\frac{1}{6c-36}$$;
5. $$\frac{y+3}{xy-y^2}-\frac{x+3}{x^2-xy}$$;
6. $$\frac{a^2+64}{2a^2+16a}+\frac{8}{a+8}$$.

1. $$\frac{4a}{7a+49}+\frac{4}{a+7}=\frac{4a}{7(a+7)}+\frac{4}{a+7}=\frac{4a+28}{7(a+7)}=\frac{4(a+7)}{7(a+7)}=\frac{4}{7}.$$

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

   $$=\frac{(a-2b)^2}{b(a-2b)}=\frac{a-2b}{b}.$$

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

   $$=\frac{6m-(4m+5)}{m(2m-5)}=\frac{2m-5}{m(2m-5)}=\frac{1}{m}.$$

4. $$\frac{1}{c^2-6c}-\frac{1}{6c-36}=\frac{1}{c(c-6)}-\frac{1}{6(c-6)}$$

   $$=\frac{6-c}{6c(c-6)}=\frac{-(c-6)}{6c(c-6)}=-\frac{1}{6c}.$$

5. $$\frac{y+3}{xy-y^2}-\frac{x+3}{x^2-xy}=\frac{y+3}{y(x-y)}-\frac{x+3}{x(x-y)}$$

   $$=\frac{x(y+3)-y(x+3)}{xy(x-y)}=\frac{xy+3x-xy-3y}{xy(x-y)}$$

   $$=\frac{3x-3y}{xy(x-y)}=\frac{3(x-y)}{xy(x-y)}=\frac{3}{xy}.$$

6. $$\frac{a^2+64}{2a^2+16a}+\frac{8}{a+8}=\frac{a^2+64}{2a(a+8)}+\frac{8}{a+8}$$

   $$=\frac{a^2+64+16a}{2a(a+8)}=\frac{(a+8)^2}{2a(a+8)}=\frac{a+8}{2a}.$$