5. Найдите а² и а, если а=√15 - 6√6-√15 + 6√6

Найдем $$a^2$$ и $$a$$, если $$a = \sqrt{15 - 6\sqrt{6}} - \sqrt{15 + 6\sqrt{6}}$$.

$$ a^2 = (\sqrt{15 - 6\sqrt{6}} - \sqrt{15 + 6\sqrt{6}})^2 = 15 - 6\sqrt{6} - 2\sqrt{(15 - 6\sqrt{6})(15 + 6\sqrt{6})} + 15 + 6\sqrt{6} = 30 - 2\sqrt{225 - 36 \cdot 6} = 30 - 2 \sqrt{225 - 216} = 30 - 2 \sqrt{9} = 30 - 2 \cdot 3 = 30 - 6 = 24 $$

$$ a = \sqrt{15 - 6\sqrt{6}} - \sqrt{15 + 6\sqrt{6}} $$

$$ a = \sqrt{9 + 6 - 2 \cdot 3 \sqrt{6}} - \sqrt{9 + 6 + 2 \cdot 3 \sqrt{6}} = \sqrt{(3 - \sqrt{6})^2} - \sqrt{(3 + \sqrt{6})^2} = |3 - \sqrt{6}| - |3 + \sqrt{6}| = 3 - \sqrt{6} - (3 + \sqrt{6}) = -2\sqrt{6} $$

Ответ: $$ a^2 = 24, a = -2\sqrt{6} $$