Упр.17.25 ГДЗ Мерзляк Поляков 8 класс (Алгебра)

1. Освободитесь от иррациональности в знаменателе дроби: 1) $$\frac{\sqrt{5}}{\sqrt{5}-2}$$; 2) $$\frac{8}{\sqrt{10}-\sqrt{2}}$$; 3) $$\frac{9}{\sqrt{x}+\sqrt{y}}$$; 4) $$\frac{2-\sqrt{2}}{2+\sqrt{2}}$$.

1. $$\frac{\sqrt{5}}{\sqrt{5}-2}=\frac{\sqrt{5}(\sqrt{5}+2)}{(\sqrt{5}-2)(\sqrt{5}+2)}=\frac{\sqrt{5}(\sqrt{5}+2)}{5-4}=\sqrt{5}(\sqrt{5}+2)=5+2\sqrt{5}.$$

2. $$\frac{8}{\sqrt{10}-\sqrt{2}}=\frac{8(\sqrt{10}+\sqrt{2})}{(\sqrt{10}-\sqrt{2})(\sqrt{10}+\sqrt{2})}=\frac{8(\sqrt{10}+\sqrt{2})}{10-2}=\frac{8(\sqrt{10}+\sqrt{2})}{8}=\sqrt{10}+\sqrt{2}.$$

3. $$\frac{9}{\sqrt{x}+\sqrt{y}}=\frac{9(\sqrt{x}-\sqrt{y})}{(\sqrt{x}+\sqrt{y})(\sqrt{x}-\sqrt{y})}=\frac{9(\sqrt{x}-\sqrt{y})}{x-y}.$$

4. $$\frac{2-\sqrt{2}}{2+\sqrt{2}}=\frac{(2-\sqrt{2})(2-\sqrt{2})}{(2+\sqrt{2})(2-\sqrt{2})}=\frac{4-4\sqrt{2}+2}{4-2}=\frac{6-4\sqrt{2}}{2}=3-2\sqrt{2}.$$

Ответ

1) $$5+2\sqrt{5}$$; 2) $$\sqrt{10}+\sqrt{2}$$; 3) $$\frac{9(\sqrt{x}-\sqrt{y})}{x-y}$$; 4) $$3-2\sqrt{2}$$.