Упр.17.8 ГДЗ Мерзляк Поляков 8 класс (Алгебра)

Выполните умножение:

1. 1) $$(\sqrt{2}+\sqrt{5})(2\sqrt{2}-\sqrt{5});$$
   2) $$(\sqrt{b}-\sqrt{c})(\sqrt{b}+\sqrt{c});$$
   3) $$(4\sqrt{2}-2\sqrt{3})(2\sqrt{3}+4\sqrt{2});$$
   4) $$(2-3\sqrt{3})^2.$$

1. $$\left(\sqrt{2}+\sqrt{5}\right)\left(2\sqrt{2}-\sqrt{5}\right)=2\sqrt{2}\cdot\sqrt{2}-\sqrt{2}\cdot\sqrt{5}+2\sqrt{2}\cdot\sqrt{5}-\sqrt{5}\cdot\sqrt{5}$$
   $$=2\cdot 2-\sqrt{10}+2\sqrt{10}-5=4+\sqrt{10}-5=\sqrt{10}-1.$$

2. $$\left(\sqrt{b}-\sqrt{c}\right)\left(\sqrt{b}+\sqrt{c}\right)=\left(\sqrt{b}\right)^2-\left(\sqrt{c}\right)^2=b-c.$$

3. $$\left(4\sqrt{2}-2\sqrt{3}\right)\left(2\sqrt{3}+4\sqrt{2}\right)=\left(4\sqrt{2}\right)^2-\left(2\sqrt{3}\right)^2$$
   $$=16\cdot 2-4\cdot 3=32-12=20.$$

4. $$\left(2-3\sqrt{3}\right)^2=2^2-2\cdot 2\cdot 3\sqrt{3}+\left(3\sqrt{3}\right)^2$$
   $$=4-12\sqrt{3}+9\cdot 3=4-12\sqrt{3}+27=31-12\sqrt{3}.$$

Ответ

1) $$\sqrt{10}-1$$; 2) $$b-c$$; 3) $$20$$; 4) $$31-12\sqrt{3}$$.