Answer
$-5+2\sqrt{6}$
Work Step by Step
Multiplying by the conjugate of the denominator, the rationalized-denominator form of the given expression, $
\dfrac{\sqrt[]{2}-\sqrt{3}}{\sqrt[]{2}+\sqrt{3}}
,$ is
\begin{array}{l}\require{cancel}
\dfrac{\sqrt[]{2}-\sqrt{3}}{\sqrt[]{2}+\sqrt{3}}\cdot\dfrac{\sqrt[]{2}-\sqrt{3}}{\sqrt[]{2}-\sqrt{3}}
\\\\=
\dfrac{(\sqrt[]{2}-\sqrt{3})^2}{(\sqrt[]{2})^2-(\sqrt{3})^2}
\\\\=
\dfrac{(\sqrt[]{2})^2+2(\sqrt{2})(-\sqrt{3})+(-\sqrt{3})^2}{2-3}
\\\\=
\dfrac{2-2\sqrt{2(3)}+3}{-1}
\\\\=
\dfrac{5-2\sqrt{6}}{-1}
\\\\=
-5+2\sqrt{6}
.\end{array}
