Упр.46 ГДЗ Алимов 10-11 класс (Алгебра)

1. Вычислить (46–47):
   1) $$\sqrt{9+\sqrt{17}}\cdot\sqrt{9-\sqrt{17}}$$;
   2) $$\left(\sqrt{3+\sqrt{5}}-\sqrt{3-\sqrt{5}}\right)^2$$;
   3) $$\left(\sqrt{5+\sqrt{21}}+\sqrt{5-\sqrt{21}}\right)^2$$.

1) $$\sqrt{9+\sqrt{17}}\cdot\sqrt{9-\sqrt{17}}=\sqrt{(9+\sqrt{17})(9-\sqrt{17})}=\sqrt{81-17}=\sqrt{64}=8.$$

2) $$\left(\sqrt{3+\sqrt5}-\sqrt{3-\sqrt5}\right)^2=(3+\sqrt5)-2\sqrt{(3+\sqrt5)(3-\sqrt5)}+(3-\sqrt5).$$

$$=6-2\sqrt{9-5}=6-2\sqrt4=6-4=2.$$

3) $$\left(\sqrt{5+\sqrt{21}}+\sqrt{5-\sqrt{21}}\right)^2=(5+\sqrt{21})+2\sqrt{(5+\sqrt{21})(5-\sqrt{21})}+(5-\sqrt{21}).$$

$$=10+2\sqrt{25-21}=10+2\sqrt4=10+4=14.$$

Ответ: 1) $$8$$; 2) $$2$$; 3) $$14$$.