$$x = 1 + \sqrt { 2 } , x - \frac { 1 } { x } =$$
$y=2$
$$y=1+\sqrt{2}-\frac{1}{1+\sqrt{2}}$$
$$y=1+\sqrt{2}-\frac{1-\sqrt{2}}{\left(1+\sqrt{2}\right)\left(1-\sqrt{2}\right)}$$
$$y=1+\sqrt{2}-\frac{1-\sqrt{2}}{1^{2}-\left(\sqrt{2}\right)^{2}}$$
$$y=1+\sqrt{2}-\frac{1-\sqrt{2}}{1-2}$$
$$y=1+\sqrt{2}-\frac{1-\sqrt{2}}{-1}$$
$$y=1+\sqrt{2}-\left(-1+\sqrt{2}\right)$$
$$y=1+\sqrt{2}+1-\sqrt{2}$$
$$y=2+\sqrt{2}-\sqrt{2}$$
$$y=2$$
$$x=1+\sqrt{2}$$ $$y=2$$
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