17. Тип 17 № 7262 i Найдите значение выражения \frac{1}{4}\left(\frac{\sqrt{2}-1}{\sqrt{2}+1}-2\frac{\sqrt{3}-1}{\sqrt{3}+1}\right)^2

Преобразуем выражение:

$$\frac{1}{4}\left(\frac{\sqrt{2}-1}{\sqrt{2}+1}-2\frac{\sqrt{3}-1}{\sqrt{3}+1}\right)^2 = \frac{1}{4}\left(\frac{(\sqrt{2}-1)^2}{(\sqrt{2}+1)(\sqrt{2}-1)}-2\frac{(\sqrt{3}-1)^2}{(\sqrt{3}+1)(\sqrt{3}-1)}\right)^2 =$$ $$\frac{1}{4}\left(\frac{2 - 2\sqrt{2} + 1}{2-1} - 2\frac{3 - 2\sqrt{3} + 1}{3-1}\right)^2 = \frac{1}{4}\left(3 - 2\sqrt{2} - 2\frac{4 - 2\sqrt{3}}{2}\right)^2 =$$ $$\frac{1}{4}\left(3 - 2\sqrt{2} - (4 - 2\sqrt{3})\right)^2 = \frac{1}{4}\left(3 - 2\sqrt{2} - 4 + 2\sqrt{3}\right)^2 = \frac{1}{4}\left(-1 - 2\sqrt{2} + 2\sqrt{3}\right)^2 =$$ $$\frac{1}{4}\left((-1 - 2\sqrt{2} + 2\sqrt{3})(-1 - 2\sqrt{2} + 2\sqrt{3})\right) = \frac{1}{4}\left(1 + 2\sqrt{2} - 2\sqrt{3} + 2\sqrt{2} + 8 - 4\sqrt{6} - 2\sqrt{3} - 4\sqrt{6} + 12\right) =$$ $$\frac{1}{4}\left(21 + 4\sqrt{2} - 4\sqrt{3} - 8\sqrt{6}\right)$$.

$$= \frac{21}{4} + \sqrt{2} - \sqrt{3} - 2\sqrt{6}$$.

Ответ: $$\frac{21}{4} + \sqrt{2} - \sqrt{3} - 2\sqrt{6}$$