i) \(\sqrt{\dfrac{4}{19-8\sqrt{3}}}-\sqrt{\dfrac{4}{19+8\sqrt{3}}}\\ =\dfrac{2}{\sqrt{19-8\sqrt{3}}}-\dfrac{2}{\sqrt{19+8\sqrt{3}}}\\ =\dfrac{2}{\sqrt{19-2\sqrt{48}}}-\dfrac{2}{\sqrt{19+2\sqrt{48}}}\\ =\dfrac{2}{\sqrt{16+2\sqrt{16}.\sqrt{3}+3}}-\dfrac{2}{\sqrt{16-2\sqrt{16}.\sqrt{3}+3}}\\ =\dfrac{2}{\sqrt{\left(4+\sqrt{3}\right)^2}}-\dfrac{2}{\sqrt{\left(4-\sqrt{3}\right)^2}}\\ =\dfrac{2}{4+\sqrt{3}}-\dfrac{2}{4-\sqrt{3}}\)
\(=\dfrac{2\left(4-\sqrt{3}\right)-2\left(4+\sqrt{3}\right)}{\left(4+\sqrt{3}\right)\left(4-\sqrt{3}\right)}\\ =\dfrac{8-2\sqrt{3}-8-2\sqrt{3}}{\left(4+\sqrt{3}\right)\left(4-\sqrt{3}\right)}\\ =\dfrac{-4\sqrt{3}}{\left(4+\sqrt{3}\right)\left(4-\sqrt{3}\right)}\)
