Aus Online Mathematik Brückenkurs 1

We start by factorizing the numbers under the root sign,

\displaystyle \begin{align} & 48=2\centerdot 24=2\centerdot 2\centerdot 12=2\centerdot 2\centerdot 2\centerdot 6=2\centerdot 2\centerdot 2\centerdot 2\centerdot 3=2^{4}\centerdot 3, \\ & 12=2\centerdot 6=2\centerdot 2\centerdot 3=2^{2}\centerdot 3, \\ & 3=3, \\ & 75=3\centerdot 25=3\centerdot 5\centerdot 5=3\centerdot 5^{2} \\ \end{align}

Now, we can take the squares out from under the root signs,

\displaystyle \begin{align} & \sqrt{48}=\sqrt{2^{4}\centerdot 3}=2^{2}\sqrt{3}=4\sqrt{3} \\ & \sqrt{12}=\sqrt{2^{2}\centerdot 3}=2\sqrt{3} \\ & \sqrt{3}=\sqrt{3} \\ & \sqrt{75}=\sqrt{3\centerdot 5^{2}}=5\sqrt{3} \\ \end{align}

and then simplify the whole expression:

\displaystyle \begin{align} & \sqrt{48}+\sqrt{12}+\sqrt{3}-\sqrt{75}=4\sqrt{3}+2\sqrt{3}+\sqrt{3}-5\sqrt{3} \\ & =\left( 4+2+1-5 \right)\sqrt{3}=2\sqrt{3} \\ \end{align}