$\dfrac{3+\sqrt[]{2}}{2\sqrt[]{2}-1}-\sqrt[]{\dfrac{1+\sqrt[]{2}}{\sqrt[]{2}-1}}$
$=\dfrac{3+\sqrt[]{2}}{2\sqrt[]{2}-1}-\sqrt[]{\dfrac{(1+\sqrt[]{2})(\sqrt[]{2}+1)}{(\sqrt[]{2}-1)(\sqrt[]{2}+1)}}$
$=\dfrac{3+\sqrt[]{2}}{2\sqrt[]{2}-1}-\sqrt[]{(\sqrt[]{2}+1)^2}$
$=\dfrac{3+\sqrt[]{2}}{2\sqrt[]{2}-1}-|\sqrt[]{2}+1|$
$=\dfrac{3+\sqrt[]{2}}{2\sqrt[]{2}-1}-(\sqrt[]{2}+1)$
$=\dfrac{3+\sqrt[]{2}-(2\sqrt[]{2}-1)(\sqrt[]{2}+1)}{2\sqrt[]{2}-1}$
$=\dfrac{3+\sqrt[]{2}-(3+\sqrt[]{2})}{2\sqrt[]{2}-1}$
$=0$
