Упр.517 ГДЗ Колягин Ткачёва 9 класс (Алгебра)

1) v(5+v21); 2) v(4+v7).

1) $$\sqrt{5+\sqrt{21}}=\sqrt{\frac72+\frac{2\sqrt{21}}4+\frac32}=\sqrt{\left(\sqrt{\frac72}\right)^2+2\sqrt{\frac72}\sqrt{\frac32}+\left(\sqrt{\frac32}\right)^2}$$

$$=\sqrt{\left(\sqrt{\frac72}+\sqrt{\frac32}\right)^2}=\sqrt{\frac72}+\sqrt{\frac32}=\frac{\sqrt7+\sqrt3}{\sqrt2}.$$

2) $$\sqrt{4+\sqrt7}=\sqrt{\frac72+\frac{2\sqrt7}4+\frac12}=\sqrt{\left(\sqrt{\frac72}\right)^2+2\sqrt{\frac72}\sqrt{\frac12}+\left(\sqrt{\frac12}\right)^2}$$

$$=\sqrt{\left(\sqrt{\frac72}+\sqrt{\frac12}\right)^2}=\sqrt{\frac72}+\sqrt{\frac12}=\frac{\sqrt7+1}{\sqrt2}.$$

Ответ

1) $$\frac{\sqrt7+\sqrt3}{\sqrt2}$$

2) $$\frac{\sqrt7+1}{\sqrt2}$$