If \(\frac{2 \sqrt{3} - \sqrt{2}}{\sqrt{3} + 2 \sqrt{2}} = m + n\sqrt{6}\),
find the values of m and n respectively
A. 1, - 2
B. - 2, n = 1
C. \(\frac{-2}{5}\), 1
D. \(\frac{2}{3}\)
Correct Answer: B
Explanation
\(\frac{2 \sqrt{3} - \sqrt{2}}{\sqrt{3} + 2 \sqrt{2}}= m + n\sqrt{6}\)
\(\frac{2 \sqrt{3} - \sqrt{2}}{\sqrt{3} + 2 \sqrt{2}}\) 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
