Упр.23.17 ГДЗ Мордкович Семенов 8 класс (Алгебра)

а) vm/(n — 2vn) : vm/(2vn — 4);
б) (vn + n)/vm · m/(6 + 6vn);
в) (n — 25)/10n : (vn + 5)/(5vn);
г) (v(nm) + 3n)/m : (3vn + vm)/vm;
д) (8vn)/(n — vn) · (3vn — 3)/(4vn);
е) (6 — vm)/vn · (9n)/(m — 36).

1. а)

   $$\frac{\sqrt{m}}{n-2\sqrt{n}}:\frac{\sqrt{m}}{2\sqrt{n}-4}= \frac{\sqrt{m}}{n-2\sqrt{n}}\cdot\frac{2\sqrt{n}-4}{\sqrt{m}}$$
   $$=\frac{2(\sqrt{n}-2)}{n-2\sqrt{n}}= \frac{2(\sqrt{n}-2)}{\sqrt{n}(\sqrt{n}-2)}= \frac{2}{\sqrt{n}}$$

2. б)

   $$\frac{\sqrt{n}+n}{\sqrt{m}}\cdot\frac{m}{6+6\sqrt{n}}= \frac{\sqrt{n}(1+\sqrt{n})\cdot m}{\sqrt{m}\cdot 6(1+\sqrt{n})}$$
   $$=\frac{\sqrt{n}\,m}{6\sqrt{m}}= \frac{\sqrt{mn}}{6}$$

3. в)

   $$\frac{n-25}{10n}:\frac{\sqrt{n}+5}{5\sqrt{n}}= \frac{(\sqrt{n}-5)(\sqrt{n}+5)}{10\sqrt{n}\sqrt{n}}\cdot\frac{5\sqrt{n}}{\sqrt{n}+5}$$
   $$=\frac{\sqrt{n}-5}{2\sqrt{n}}$$

4. г)

   $$\frac{\sqrt{nm}+3n}{m}:\frac{3\sqrt{n}+\sqrt{m}}{\sqrt{m}}= \frac{\sqrt{n}(\sqrt{m}+3\sqrt{n})\cdot\sqrt{m}}{\sqrt{m}\sqrt{m}\cdot(3\sqrt{n}+\sqrt{m})}$$
   $$=\frac{\sqrt{n}}{\sqrt{m}}$$

5. д)

   $$\frac{8\sqrt{n}}{n-\sqrt{n}}\cdot\frac{3\sqrt{n}-3}{4\sqrt{n}}= \frac{8\sqrt{n}\cdot 3(\sqrt{n}-1)}{\sqrt{n}(\sqrt{n}-1)\cdot 4\sqrt{n}}$$
   $$=\frac{6}{\sqrt{n}}$$

6. е)

   $$\frac{6-\sqrt{m}}{\sqrt{n}}\cdot\frac{9n}{m-36}= \frac{(6-\sqrt{m})\cdot 9\sqrt{n}\sqrt{n}}{\sqrt{n}(\sqrt{m}-6)(\sqrt{m}+6)}$$
   $$=\frac{9\sqrt{n}}{\sqrt{m}+6}$$

Ответ

а) $$\frac{2}{\sqrt{n}}$$; б) $$\frac{\sqrt{mn}}{6}$$; в) $$\frac{\sqrt{n}-5}{2\sqrt{n}}$$; г) $$\frac{\sqrt{n}}{\sqrt{m}}$$; д) $$\frac{6}{\sqrt{n}}$$; е) $$\frac{9\sqrt{n}}{\sqrt{m}+6}$$.