4. Найдите значение выражения a+2b a²-2ab b : 2b-a при а = 1,6, b = √2-1.

Упростим выражение:

$$ \frac{a+2b}{a^2 - 2ab} : \frac{b}{2b-a} = \frac{a+2b}{a(a-2b)} \cdot \frac{2b-a}{b} = -\frac{a+2b}{ab} $$

Подставим значения a и b:

$$ -\frac{1,6 + 2(\sqrt{2}-1)}{1,6(\sqrt{2}-1)} = -\frac{1,6 + 2\sqrt{2}-2}{1,6(\sqrt{2}-1)} = -\frac{2\sqrt{2}-0,4}{1,6(\sqrt{2}-1)} = -\frac{2(\sqrt{2}-0,2)}{1,6(\sqrt{2}-1)} = -\frac{2(\sqrt{2}-0,2)}{1,6(\sqrt{2}-1)} $$

$$ = -\frac{5(\sqrt{2}-0,2)}{4(\sqrt{2}-1)} = -\frac{5(\sqrt{2}-0,2)(\sqrt{2}+1)}{4(\sqrt{2}-1)(\sqrt{2}+1)} = -\frac{5(2 + \sqrt{2} - 0,2\sqrt{2} - 0,2)}{4(2-1)} = -\frac{5(1,8+0,8\sqrt{2})}{4} = -\frac{9+4\sqrt{2}}{4} = -2,25 - \sqrt{2} \approx -3,66 $$

Ответ: -3,66