[tex]\it a = \sqrt{\sqrt3+\sqrt{19-8\sqrt3}} \ \ \ \ \ (1)
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\sqrt{19-8\sqrt3} = \sqrt{16-8\sqrt3+3} = \sqrt{(4-\sqrt3)^2} =4-\sqrt3 \ \ \ \ (2)
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(1), (2) \Rightarrow a = \sqrt{\sqrt3+4-\sqrt3} = \sqrt4 = 2 \in \mathbb{N}[/tex]
