Question

$\sqrt{2 + \sqrt{6} }$
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Answer 1

$we \: know \: that \\ \sqrt{m + n} \\ = \sqrt{ \frac{m + \sqrt{ {m}^{2} - {n}^{2} } }{2} } + \sqrt{ \frac{m - \sqrt{ {m}^{2} - {n}^{2} } }{2} } \\ so \\ \sqrt{2 + \sqrt{6} } \\ m = 2 \: \: \: \: \: \: \: n = \sqrt{6} \\ = \sqrt{ \frac{2 + \sqrt{2 {}^{2} - \sqrt{6} {}^{2} } }{2} } + \sqrt{ \frac{2 - \sqrt{2 {}^{2} - { \sqrt{6} }^{2} } }{2} } \\ = \sqrt{ \frac{2 + \sqrt{4 - 6} }{2} } + \sqrt{ \frac{2 - \sqrt{4 - 6} }{2} } \\ = \sqrt{ \frac{2 + \sqrt{2} i}{2} } + \sqrt{ \frac{2 - \sqrt{2}i }{2} }$

Answer 2

Step-by-step explanation:

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