Упр.176 ГДЗ Мерзляк Полонский 8 класс (Алгебра)

1) (a+2)/(a2-2a+1) : (a2-4)/(3a-3) — 3/(a-2);
2) (b2+3b)/(b3+9b) * ((b-3)/(b+3) + (b+3)/(b-3)); Упростите выражение:
1) (a/3 + a/4) * 6/a2;
2) a2b/(a-b) * (1/b — 1/a);

1. $$\frac{a+2}{a^2-2a+1}:\frac{a^2-4}{3a-3}-\frac{3}{a-2}$$

   $$\frac{a+2}{(a-1)^2}\cdot\frac{3(a-1)}{(a-2)(a+2)}-\frac{3}{a-2}=\frac{3}{(a-1)(a-2)}-\frac{3}{a-2}$$

   $$\frac{3-3(a-1)}{(a-1)(a-2)}=\frac{6-3a}{(a-1)(a-2)}=-\frac{3(a-2)}{(a-1)(a-2)}=\frac{3}{1-a}$$

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

   $$\frac{b(b+3)}{b(b^2+9)}\cdot\frac{(b-3)^2+(b+3)^2}{(b-3)(b+3)}$$

   $$\frac{1}{b^2+9}\cdot\frac{2b^2+18}{b-3}=\frac{2}{b-3}$$

3. $$\left(\frac{3c+1}{3c-1}-\frac{3c-1}{3c+1}\right):\frac{2c}{6c+2}$$

   $$\frac{(3c+1)^2-(3c-1)^2}{(3c-1)(3c+1)}\cdot\frac{2(3c+1)}{2c}$$

   $$\frac{12c}{3c-1}\cdot\frac{1}{c}=\frac{12}{3c-1}$$

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

   $$\left(\frac{1}{(a-2b)^2}+\frac{1}{(a-2b)(a+2b)}\right)\cdot\frac{(a-2b)(a+2b)}{2a}$$

   $$\frac{a+2b+a-2b}{(a-2b)(a+2b)}\cdot\frac{(a-2b)(a+2b)}{2a}=\frac{1}{a-2b}$$

5. $$\left(\frac{a-8}{a^2-10a+25}-\frac{a}{a^2-25}\right):\frac{a-20}{(a-5)^2}$$

   $$\left(\frac{a-8}{(a-5)^2}-\frac{a}{(a-5)(a+5)}\right)\cdot\frac{(a-5)^2}{a-20}$$

   $$\frac{(a-8)(a+5)-a(a-5)}{(a-5)^2(a+5)}\cdot\frac{(a-5)^2}{a-20}=\frac{2(a-20)}{a+5}\cdot\frac{1}{a-20}=\frac{2}{a+5}$$

6. $$\left(\frac{2x+1}{x^2+6x+9}-\frac{x-2}{x^2+3x}\right):\frac{x^2+6}{x^3-9x}$$

   $$\left(\frac{2x+1}{(x+3)^2}-\frac{x-2}{x(x+3)}\right)\cdot\frac{x(x^2-9)}{x^2+6}$$

   $$\frac{x(2x+1)-(x-2)(x+3)}{x(x+3)^2}\cdot\frac{x(x-3)(x+3)}{x^2+6}$$

   $$\frac{x^2+6}{x+3}\cdot\frac{x-3}{x^2+6}=\frac{x-3}{x+3}$$

Упростите выражение

1. $$\left(\frac{a}{3}+\frac{a}{4}\right)\cdot\frac{6}{a^2}=\frac{7a}{12}\cdot\frac{6}{a^2}=\frac{7}{2a}$$

2. $$\frac{a^2b}{a-b}\cdot\left(\frac{1}{b}-\frac{1}{a}\right)=\frac{a^2b}{a-b}\cdot\frac{a-b}{ab}=a$$

3. $$\left(1+\frac{a}{b}\right):\left(1-\frac{a}{b}\right)=\frac{b+a}{b}:\frac{b-a}{b}=\frac{b+a}{b-a}$$

4. $$\left(\frac{a^2}{b^2}-\frac{2a}{b}+1\right)\cdot\frac{b}{a-b}=\frac{a^2-2ab+b^2}{b^2}\cdot\frac{b}{a-b}$$

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

5. $$\frac{a^2-ab}{b^2-1}:\frac{a}{b}-\frac{a}{b-1}$$

   $$\frac{a(a-b)}{(b-1)(b+1)}\cdot\frac{b}{a}-\frac{a}{b-1}=\frac{b(a-b)}{(b-1)(b+1)}-\frac{a}{b-1}$$

   $$\frac{a-b}{b-1}-\frac{a}{b-1}=\frac{-b}{b-1}=\frac{b}{1-b}$$

6. $$\left(\frac{5}{m-n}-\frac{4}{m+n}\right):\frac{m+9n}{m+n}$$

   $$\frac{5(m+n)-4(m-n)}{(m-n)(m+n)}\cdot\frac{m+n}{m+9n}=\frac{m+9n}{m-n}\cdot\frac{1}{m+9n}=\frac{1}{m-n}$$

7. $$\frac{x-2}{x+2}\cdot\left(x-\frac{x^2}{x-2}\right)$$

   $$\frac{x-2}{x+2}\cdot\frac{x(x-2)-x^2}{x-2}=\frac{x-2}{x+2}\cdot\frac{-2x}{x-2}=-\frac{2x}{x+2}$$

8. $$\frac{x^2+x}{4}:\frac{x^2}{4}+\frac{x-1}{x}$$

   $$\frac{x(x+1)}{4}\cdot\frac{4}{x^2}+\frac{x-1}{x}=\frac{x+1}{x}+\frac{x-1}{x}=\frac{2x}{x}=2$$

9. $$\frac{6c^2}{c^2-1}:\left(\frac{1}{c-1}+1\right)$$

   $$\frac{6c^2}{(c-1)(c+1)}:\frac{c}{c-1}=\frac{6c^2}{(c-1)(c+1)}\cdot\frac{c-1}{c}=\frac{6c}{c+1}$$

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

    $$\frac{x(x-y)+y(x+y)}{(x+y)(x-y)}\cdot\frac{x(x+y)}{x^2+y^2}$$

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

Ответ

1) $$\frac{3}{1-a}$$; 2) $$\frac{2}{b-3}$$; 3) $$\frac{12}{3c-1}$$; 4) $$\frac{1}{a-2b}$$; 5) $$\frac{2}{a+5}$$; 6) $$\frac{x-3}{x+3}$$.

$$\frac{7}{2a};\ a;\ \frac{b+a}{b-a};\ \frac{a-b}{b};\ \frac{b}{1-b};\ \frac{1}{m-n};\ -\frac{2x}{x+2};\ 2;\ \frac{6c}{c+1};\ \frac{x}{x-y}$$