If 2√3−√2√3+2√2 2 3 − 2 3 + 2 2 = m + n √ 6, find the values of m and n respectively.

If $\frac{2\sqrt{3}-\sqrt{2}}{\sqrt{3}+2\sqrt{2}}$ = m + n √ 6, find the values of m and n respectively.

Answer Details

$\frac{2\sqrt{3}-\sqrt{2}}{\sqrt{3}+2\sqrt{2}}$= m + n√6
$\frac{2\sqrt{3}-\sqrt{2}}{\sqrt{3}+2\sqrt{2}}$ x $\frac{\sqrt{3}-2\sqrt{2}}{\sqrt{3}-\sqrt{2}}$

$\frac{2\sqrt{3}(\sqrt{3}-2\sqrt{2})-\sqrt{2}(\sqrt{3}-2\sqrt{2})}{\sqrt{3}(\sqrt{3}-2\sqrt{2})+2\sqrt{2}(\sqrt{3}-2\sqrt{2})}$
$\frac{2\times 3-4\sqrt{6}-6+2\times 2}{3-2\sqrt{6}+2\sqrt{6}-4\times 2}$
= $\frac{6-4\sqrt{6}-\sqrt{6}+4}{3-8}$
= $\frac{0-4\sqrt{6}-6}{5}$
= $\frac{10-5\sqrt{6}}{5}$
= − 2 + √6
∴ m + n$\sqrt{6}$ = − 2 + √6
m = − 2, n = 1