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$\sqrt{-\sqrt{3} +\sqrt{3+8\sqrt{7+4\sqrt{3}}}}\\ =\sqrt{-\sqrt{3} +\sqrt{3+8\sqrt{(2+\sqrt{3})^2}}} \\ =\sqrt{-\sqrt{3} +\sqrt{3+8\sqrt{3}+16}}\\ =\sqrt{-\sqrt{3} +\sqrt{(4+\sqrt{3})^2}} \\ =\sqrt{-\sqrt{3} +4+\sqrt{3}}\\ =\sqrt4 =2$
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