Контрольная работа 2 Вариант 2 Дидактические материалы ГДЗ Мерзляк Полонский 8 класс (Алгебра)

1) $$-\frac{54a^6b^9}{c^{12}}\cdot\left(-\frac{c^{20}}{12a^4b^{15}}\right)=\frac{54a^6b^9\cdot c^{20}}{12a^4b^{15}c^{12}}=\frac{9a^2c^8}{2b^6}.$$

2) $$\frac{98m^8}{p^{17}}:\left(49m^5p^2\right)=\frac{98m^8}{p^{17}\cdot49m^5p^2}=\frac{2m^3}{p^{19}}.$$

3) $$\frac{5a+5b}{b}\cdot\frac{6b^2}{a^2-b^2}=\frac{5(a+b)\cdot6b^2}{b(a-b)(a+b)}=\frac{30b}{a-b}.$$

4) $$\frac{x^2-49}{3x-24}:\frac{5x+35}{x-8}=\frac{(x-7)(x+7)}{3(x-8)}\cdot\frac{x-8}{5(x+7)}=\frac{x-7}{15}.$$

1) $$\frac{3a}{a-3}+\frac{a+5}{6-2a}\cdot\frac{54}{5a+a^2}$$

$$=\frac{3a}{a-3}+\frac{a+5}{2(3-a)}\cdot\frac{54}{a(a+5)}$$

$$=\frac{3a}{a-3}+\frac{27}{a(3-a)}=\frac{3a}{a-3}-\frac{27}{a(a-3)}$$

$$=\frac{3a^2-27}{a(a-3)}=\frac{3(a^2-9)}{a(a-3)}=\frac{3(a-3)(a+3)}{a(a-3)}=\frac{3(a+3)}{a}.$$

2) $$\left(\frac{a+4}{a-4}-\frac{a-4}{a+4}\right):\frac{48a}{16-a^2}$$

$$=\frac{(a+4)^2-(a-4)^2}{(a-4)(a+4)}:\frac{48a}{16-a^2}$$

$$=\frac{16a}{a^2-16}\cdot\frac{16-a^2}{48a}=-\frac{1}{3}.$$

3) $$\left(\frac{a^2}{a+5}-\frac{a^3}{a^2+10a+25}\right):\left(\frac{a}{a+5}-\frac{a^2}{a^2-25}\right)$$

$$=\left(\frac{a^2}{a+5}-\frac{a^3}{(a+5)^2}\right):\left(\frac{a}{a+5}-\frac{a^2}{(a-5)(a+5)}\right)$$

$$=\frac{a^2(a+5)-a^3}{(a+5)^2}:\frac{a(a-5)-a^2}{(a-5)(a+5)}$$

$$=\frac{a^2\cdot5}{(a+5)^2}:\frac{-5a}{(a-5)(a+5)}$$

$$=\frac{5a^2}{(a+5)^2}\cdot\frac{(a-5)(a+5)}{-5a}=\frac{a(a-5)}{-(a+5)}=\frac{5a-a^2}{a+5}.$$

4) Из условия $$x^2+\frac{49}{x^2}=50.$$ Тогда

$$x^2-2\cdot x\cdot\frac{7}{x}+\frac{49}{x^2}=50-14,$$

$$\left(x-\frac{7}{x}\right)^2=36,$$

$$x-\frac{7}{x}=\pm6.$$

Ответ

1) $$\frac{9a^2c^8}{2b^6};$$ 2) $$\frac{2m^3}{p^{19}};$$ 3) $$\frac{30b}{a-b};$$ 4) $$\frac{x-7}{15};$$ 2.1) $$\frac{3(a+3)}{a};$$ 2.2) $$-\frac{1}{3};$$ 3) тождество верно; 4) $$x-\frac{7}{x}=\pm6.$$