Question

Simplify: $\frac{4+\sqrt{3}+5\sqrt{2}}{\sqrt{48}+\sqrt{18}}$

Answer 1

$\frac{4+\sqrt{3}+5\sqrt{2}}{\sqrt{2\times 2\times 2\times 2\times 3}+\sqrt{2\times 3\times 3}}\Rightarrow \frac{4+\sqrt{3}+5\sqrt{2}}{4\sqrt{3}+3\sqrt{2}}\Rightarrow \frac{(4+\sqrt{3}+5\sqrt{2})(4\sqrt{3}-3\sqrt{2})}{{\left(4\sqrt{3}\right)}^2-{\left(3\sqrt{2}\right)}^2}\Rightarrow \frac{(4+\sqrt{3}+5\sqrt{2})(4\sqrt{3}-3\sqrt{2})}{48-18}\Rightarrow \frac{16\sqrt{3}-12\sqrt{2}+12-3\sqrt{6}+20\sqrt{6}-30}{30}\Rightarrow \frac{16\sqrt{3}-12\sqrt{2}-17\sqrt{6}-18}{30}$