\(\frac{2 \sqrt{3} - \sqrt{2}}{\sqrt{3} + 2 \sqrt{2}}= m + n\sqrt{6}\)

\(\frac{2 \sqrt{3} - \sqrt{2}}{\sqrt{3} + 2 \sqrt{2}}\) 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