Aus Online Mathematik Brückenkurs 1

The trick is to use the conjugate rule \displaystyle \left( a-b \right)(a+b)=a^{\text{2}}-b^{\text{2}} and multiply the top and bottom of the fraction by \displaystyle 3-\sqrt{7} (note the minus sign), since then the new denominator will be \displaystyle \left( 3+\sqrt{7} \right)\left( 3-\sqrt{7} \right)=3^{2}-\left( \sqrt{7} \right)^{2}=9-7=2 (conjugate rule with \displaystyle a=\text{3 } and \displaystyle b=\sqrt{\text{7}} ), i.e. the root sign is squared away.

The whole calculation is

\displaystyle \begin{align} & \frac{2}{3+\sqrt{7}}=\frac{2}{3+\sqrt{7}}\centerdot \frac{3-\sqrt{7}}{3-\sqrt{7}}=\frac{2\left( 3-\sqrt{7} \right)}{3^{2}-\left( \sqrt{7} \right)^{2}} \\ & =\frac{2\centerdot 3-2\sqrt{7}}{2}=3-\sqrt{7} \\ \end{align}