4\sqrt3 + \sqrt27=4\sqrt3 +\sqrt{3*3^2}=\sqrt3(4+3)= 7\sqrt3
\frac{\sqrt2 + 3\sqrt8 - \sqrt18}{\sqrt2 }=\frac{\sqrt2 + 3\sqrt{2*2^2} - \sqrt{2*3^2}}{\sqrt2 }=\frac{\sqrt2 + 3*2\sqrt2 - 3\sqrt2}{\sqrt2 }=\frac{\sqrt2(1+6-3)}{\sqrt2 }=4
6\sqrt{50} + \sqrt{32} - 3\sqrt{72}=6\sqrt{{2*5^2}} + \sqrt{{2*4^2}} - 3\sqrt{{2*6^2}}=\sqrt{2}(6*5+4-3*6)=16\sqrt{2}
\frac{\sqrt{24} - \sqrt{54}}{\sqrt6}=\frac{\sqrt{6*2^2} - \sqrt{6*3^2}}{\sqrt6}=\frac{\sqrt{6}(2-3)}{\sqrt6}=-1