15.86 ( 6+4/2 √2 + √6+4√2+ 6-4/2 √2-√6-4√2 )2 = 8.

$$\left(\frac{6+4\sqrt{2}}{\sqrt{2} + \sqrt{6+4\sqrt{2}}} + \frac{6-4\sqrt{2}}{\sqrt{2} - \sqrt{6-4\sqrt{2}}}\right)^2 = 8$$

Сначала упростим $$\sqrt{6+4\sqrt{2}} = \sqrt{2 + 4\sqrt{2} + 4} = \sqrt{(\sqrt{2}+2)^2} = 2 + \sqrt{2}$$

Аналогично, $$\sqrt{6-4\sqrt{2}} = \sqrt{2 - 4\sqrt{2} + 4} = \sqrt{(2-\sqrt{2})^2} = 2 - \sqrt{2}$$

Тогда $$\left(\frac{6+4\sqrt{2}}{\sqrt{2} + 2 + \sqrt{2}} + \frac{6-4\sqrt{2}}{\sqrt{2} - (2 - \sqrt{2})}\right)^2 = \left(\frac{6+4\sqrt{2}}{2\sqrt{2} + 2} + \frac{6-4\sqrt{2}}{2\sqrt{2} - 2}\right)^2 = \left(\frac{6+4\sqrt{2}}{2(\sqrt{2} + 1)} + \frac{6-4\sqrt{2}}{2(\sqrt{2} - 1)}\right)^2 = \left(\frac{(6+4\sqrt{2})(\sqrt{2}-1) + (6-4\sqrt{2})(\sqrt{2}+1)}{2((\sqrt{2}+1)(\sqrt{2}-1))}\right)^2 = \left(\frac{6\sqrt{2} - 6 + 8 - 4\sqrt{2} + 6\sqrt{2} + 6 - 8 - 4\sqrt{2}}{2(2 - 1)}\right)^2 = \left(\frac{4\sqrt{2}}{2}\right)^2 = (2\sqrt{2})^2 = 4 \cdot 2 = 8$$

Ответ: Равенство верно