Упр.23.1 ГДЗ Мордкович Семенов 8 класс (Алгебра)

а) (6v75 — v125) · v3;
б) (11v45 — 4v80) : 3v5;
в) (6 + v42)(v7 — v6);
г) (5v32 — 4v18) · v2;
д) (7v63 + 3v112) : 3v7;
е) (5 + v35)(v5 — v7).

1. $$ (6\sqrt{75}-\sqrt{125})\cdot \sqrt{3} = (6\sqrt{25\cdot 3}-\sqrt{25\cdot 5})\cdot \sqrt{3} = (6\cdot 5\sqrt{3}-5\sqrt{5})\cdot \sqrt{3} $$

   $$ = (30\sqrt{3}-5\sqrt{5})\cdot \sqrt{3} = 30\cdot 3-5\sqrt{15} = 90-5\sqrt{15} $$

2. $$ (11\sqrt{45}-4\sqrt{80}) : 3\sqrt{5} = (11\sqrt{9\cdot 5}-4\sqrt{16\cdot 5}) : 3\sqrt{5} $$

   $$ = (11\cdot 3\sqrt{5}-4\cdot 4\sqrt{5}) : 3\sqrt{5} = (33\sqrt{5}-16\sqrt{5}) : 3\sqrt{5} $$

   $$ = \frac{17\sqrt{5}}{3\sqrt{5}} = \frac{17}{3} = 5\frac{2}{3} $$

3. $$ (6+\sqrt{42})(\sqrt{7}-\sqrt{6}) = 6\sqrt{7}-6\sqrt{6}+\sqrt{42}\cdot \sqrt{7}-\sqrt{42}\cdot \sqrt{6} $$

   $$ = 6\sqrt{7}-6\sqrt{6}+\sqrt{42\cdot 7}-\sqrt{42\cdot 6} = 6\sqrt{7}-6\sqrt{6}+\sqrt{7\cdot 6\cdot 7}-\sqrt{7\cdot 6\cdot 6} $$

   $$ = 6\sqrt{7}-6\sqrt{6}+7\sqrt{6}-6\sqrt{7} = \sqrt{6} $$

4. $$ (5\sqrt{32}-4\sqrt{18})\cdot \sqrt{2} = (5\sqrt{16\cdot 2}-4\sqrt{9\cdot 2})\cdot \sqrt{2} $$

   $$ = (5\cdot 4\sqrt{2}-4\cdot 3\sqrt{2})\cdot \sqrt{2} = (20\sqrt{2}-12\sqrt{2})\cdot \sqrt{2} $$

   $$ = 8\sqrt{2}\cdot \sqrt{2} = 8\cdot 2 = 16 $$

5. $$ (7\sqrt{63}+3\sqrt{112}) : 3\sqrt{7} = (7\sqrt{9\cdot 7}+3\sqrt{16\cdot 7}) : 3\sqrt{7} $$

   $$ = (7\cdot 3\sqrt{7}+3\cdot 4\sqrt{7}) : 3\sqrt{7} = (21\sqrt{7}+12\sqrt{7}) : 3\sqrt{7} $$

   $$ = \frac{33\sqrt{7}}{3\sqrt{7}} = 11 $$

6. $$ (5+\sqrt{35})(\sqrt{5}-\sqrt{7}) = 5\sqrt{5}-5\sqrt{7}+\sqrt{35}\cdot \sqrt{5}-\sqrt{35}\cdot \sqrt{7} $$

   $$ = 5\sqrt{5}-5\sqrt{7}+\sqrt{35\cdot 5}-\sqrt{35\cdot 7} = 5\sqrt{5}-5\sqrt{7}+\sqrt{5\cdot 7\cdot 5}-\sqrt{5\cdot 7\cdot 7} $$

   $$ = 5\sqrt{5}-5\sqrt{7}+5\sqrt{7}-7\sqrt{5} = -2\sqrt{5} $$

Ответ

а) $$90-5\sqrt{15}$$; б) $$\frac{17}{3}$$; в) $$\sqrt{6}$$; г) $$16$$; д) $$11$$; е) $$-2\sqrt{5}$$.