Упр.102 Вариант 3 Дидактические материалы ГДЗ Мерзляк Полонский 8 класс (Алгебра)

1. Упростите выражение:
   1) $$(4\sqrt{7}+7\sqrt{12}-2\sqrt{192})\cdot\sqrt{3}-\sqrt{84};$$
   2) $$(2\sqrt{5}-\sqrt{15})(\sqrt{15}+2\sqrt{5})-(\sqrt{10}-5\sqrt{2})^2;$$
   3) $$(8-\sqrt{6})^2+(5+\sqrt{6})^2;$$
   4) $$\left(\sqrt{8+2\sqrt{7}}+\sqrt{8-2\sqrt{7}}\right)^2.$$

1. $$\left(4\sqrt7+7\sqrt{12}-2\sqrt{192}\right)\cdot\sqrt3-\sqrt{84}$$
   $$=\left(4\sqrt7+7\cdot2\sqrt3-2\cdot8\sqrt3\right)\cdot\sqrt3-2\sqrt{21}$$
   $$=\left(4\sqrt7+14\sqrt3-16\sqrt3\right)\cdot\sqrt3-2\sqrt{21}$$
   $$=\left(4\sqrt7-2\sqrt3\right)\cdot\sqrt3-2\sqrt{21}$$
   $$=4\sqrt{21}-2\cdot3-2\sqrt{21}=2\sqrt{21}-6.$$

2. $$\left(2\sqrt5-\sqrt{15}\right)\left(\sqrt{15}+2\sqrt5\right)-\left(\sqrt{10}-5\sqrt2\right)^2$$
   $$=\left(2\sqrt5\right)^2-\left(\sqrt{15}\right)^2-\left(10-10\sqrt{20}+25\cdot2\right)$$
   $$=4\cdot5-15-\left(10-10\sqrt{20}+50\right)$$
   $$=20-15-60+10\sqrt{20}$$
   $$=5-60+10\sqrt{4\cdot5}$$
   $$=5-60+20\sqrt5=20\sqrt5-55.$$

3. $$\left(8-\sqrt6\right)^2+\left(5+\sqrt6\right)^2$$
   $$=64-16\sqrt6+6+25+10\sqrt6+6$$
   $$=101-6\sqrt6.$$

4. $$\left(\sqrt{8+2\sqrt7}+\sqrt{8-2\sqrt7}\right)^2$$
   $$=\left(\sqrt{8+2\sqrt7}\right)^2+2\sqrt{(8+2\sqrt7)(8-2\sqrt7)}+\left(\sqrt{8-2\sqrt7}\right)^2$$
   $$=8+2\sqrt7+2\sqrt{64-4\cdot7}+8-2\sqrt7$$
   $$=16+2\sqrt{36}=16+12=28.$$

Ответ

1) $$2\sqrt{21}-6$$; 2) $$20\sqrt5-55$$; 3) $$101-6\sqrt6$$; 4) $$28$$.