$I=\dfrac{\sqrt[]{2+\sqrt[]3}-\sqrt[]{2-\sqrt[]3}}{\sqrt[]{2+\sqrt[]3}+\sqrt[]{2-\sqrt[]3}}$
$=\dfrac{\sqrt[]{2+\sqrt[]3}-\sqrt[]{2-\sqrt[]3}}{\sqrt[]{2+\sqrt[]3}+\sqrt[]{2-\sqrt[]3}}$
$=\dfrac{(\sqrt[]{2+\sqrt[]3}-\sqrt[]{2-\sqrt[]3})^2}{(\sqrt[]{2+\sqrt[]3}-\sqrt[]{2-\sqrt[]3})(\sqrt[]{2+\sqrt[]3}+\sqrt[]{2-\sqrt[]3}}$
$=\dfrac{2+\sqrt[]3-2.\sqrt[]{2+\sqrt[]3}.\sqrt[]{2-\sqrt[]3}+2-\sqrt[]3}{2+\sqrt[]3-2+\sqrt[]3}$
$=\dfrac{4-2.\sqrt[]{4-3}}{2.\sqrt[]3}$
$=\dfrac{2}{2.\sqrt[]3}=\dfrac{1}{\sqrt[]3}=K$
