6−7+27+435=6−7+27+2⋅27⋅5=6−7+(25)2+(7)2+2⋅257=6−7+(25+7)2=6−7+25+7=(5)2+(1)2+2⋅5⋅1=(5+1)2=5+1

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$\sqrt{6 - \sqrt{7} + \sqrt{27 + 4 \sqrt{35}}} =$

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$\sqrt{5} - 1$

$\sqrt{5} - \sqrt{2}$

$\sqrt{5} + 1$

$2 + \sqrt{5}$

answer is C.

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Detailed Solution

$\begin{array}{l} \sqrt{6 - \sqrt{7} + \sqrt{27 + 4 \sqrt{35}}} \\ = \sqrt{6 - \sqrt{7} + \sqrt{27 + 2 \cdot 2 \sqrt{7} \cdot \sqrt{5}}} \\ = \sqrt{6 - \sqrt{7} + \sqrt{(2 \sqrt{5})^{2} + (\sqrt{7})^{2} + 2 \cdot 2 \sqrt{5} \sqrt{7}}} \\ = \sqrt{6 - \sqrt{7} + \sqrt{(2 \sqrt{5} + \sqrt{7})^{2}}} \end{array}$
$\begin{matrix} = \sqrt{6 - \sqrt{7} + 2 \sqrt{5} + \sqrt{7}} \\ = \sqrt{(\sqrt{5})^{2} + (1)^{2} + 2 \cdot \sqrt{5} \cdot 1} \\ = \sqrt{(\sqrt{5} + 1)^{2}} \\ = \sqrt{5} + 1 \end{matrix}$