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[tex]\large\boxed{\begin{array}{l}\begin{array}{c|c}\rm48&\rm2\\\rm24&\rm2\\\rm12&\rm2\\\rm6&\rm2\\\rm3&\rm1\end{array}\\\rm 48=2^4\cdot3\\\rm 24=2^3\cdot3\\\rm 12=2^2\cdot3\\\begin{array}{c|c}\rm75&\rm3\\\rm25&\rm5\\\rm5&\rm5\\\rm1\end{array}\\\rm 75=3\cdot5^2\end{array}}[/tex]
[tex]\large\boxed{\begin{array}{l}\rm\dfrac{2\sqrt{12}+2\sqrt{75}}{\sqrt{48}}=\dfrac{2\cdot\sqrt{2^2\cdot3}+2\sqrt{3\cdot5^2}}{\sqrt{2^4\cdot3}}\\\\\rm\dfrac{2\sqrt{12}+2\sqrt{75}}{\sqrt{48}}=\dfrac{4\sqrt{3}+10\sqrt{3}}{2^2\sqrt{3}}\\\\\rm\dfrac{2\sqrt{12}+2\sqrt{75}}{\sqrt{48}}=\dfrac{14\,\diagup\!\!\!\!\!\sqrt{\!3}}{4\,\diagup\!\!\!\!\!\!\sqrt{\!3}}\\\\\rm\dfrac{2\sqrt{12}+2\sqrt{75}}{\sqrt{48}}=\dfrac{14\div2}{4\div2}\\\\\rm\dfrac{2\sqrt{12}+2\sqrt{75}}{\sqrt{48}}=\dfrac{7}{2}\end{array}}[/tex]