Aus Online Mathematik Brückenkurs 1

We can multiply \displaystyle \sqrt{\frac{2}{3}} into the bracket and then write the root expressions together under a common root sign using the rule \displaystyle \sqrt{a}\centerdot \sqrt{b}=\sqrt{ab}

\displaystyle \sqrt{\frac{2}{3}}\left( \sqrt{6}-\sqrt{3} \right)=\sqrt{\frac{2}{3}}\centerdot \sqrt{6}-\sqrt{\frac{2}{3}}\centerdot \sqrt{3}=\sqrt{\frac{2\centerdot 6}{3}}-\sqrt{\frac{2\centerdot 3}{3}}.

Because \displaystyle \frac{2\centerdot 6}{3}=2\centerdot 2=2^{2} and \displaystyle \frac{2\centerdot 3}{3}=2, we obtain

\displaystyle \sqrt{\frac{2}{3}}\left( \sqrt{6}-\sqrt{3} \right)=\sqrt{2^{2}}-\sqrt{2}=2-\sqrt{2}