Решение:

1) $$9^{\frac{2}{3}} : 9^{\frac{1}{6}} = 9^{\frac{2}{3} - \frac{1}{6}} = 9^{\frac{4}{6} - \frac{1}{6}} = 9^{\frac{3}{6}} = 9^{\frac{1}{2}} = \sqrt{9} = 3$$

Ответ: 3

2) $$7^{\frac{2}{3}} \cdot 49^{\frac{2}{3}} = 7^{\frac{2}{3}} \cdot (7^2)^{\frac{2}{3}} = 7^{\frac{2}{3}} \cdot 7^{\frac{4}{3}} = 7^{\frac{2}{3} + \frac{4}{3}} = 7^{\frac{6}{3}} = 7^2 = 49$$

Ответ: 49

3) $$2^{1-2\sqrt{2}} \cdot 4^{\sqrt{2}} = 2^{1-2\sqrt{2}} \cdot (2^2)^{\sqrt{2}} = 2^{1-2\sqrt{2}} \cdot 2^{2\sqrt{2}} = 2^{1-2\sqrt{2} + 2\sqrt{2}} = 2^1 = 2$$

Ответ: 2