If \(\frac{(2\sqrt{3}-\sqrt{2})}{(\sqrt{3}+2\sqrt{2})} = m +n\sqrt{6}\), find the values of m and n respectively.
The correct answer is B. -2, 1
To simplify the given fraction \(\frac{2\sqrt{3} - \sqrt{2}}{\sqrt{3} + 2\sqrt{2}}\), we can use the conjugate rule to rationalize the denominator. The conjugate of the denominator \(\sqrt{3} + 2\sqrt{2}\) is \(\sqrt{3} - 2\sqrt{2}\).
Multiply both the numerator and denominator by the conjugate of the denominator:
\[\frac{2\sqrt{3} - \sqrt{2}}{\sqrt{3} + 2\sqrt{2}} \cdot \frac{\sqrt{3} - 2\sqrt{2}}{\sqrt{3} - 2\sqrt{2}}\]
Simplify each term:
\[\frac{(2\sqrt{3})(\sqrt{3}) - (2\sqrt{3})(2\sqrt{2}) - (\sqrt{2})(\sqrt{3}) + (\sqrt{2})(2\sqrt{2})}{(\sqrt{3})(\sqrt{3}) - (2\sqrt{2})(2\sqrt{2})}\]
\[\frac{6 - 4\sqrt{6} - \sqrt{6} + 4}{3 - 8}\]
\[\frac{10 - 5\sqrt{6}}{-5}\]
Divide both the numerator and denominator by -5:
\[\frac{-10 + 5\sqrt{6}}{5}\]
Now, we can separate this fraction into the sum of two terms:
\[\frac{-10}{5} + \frac{5\sqrt{6}}{5}\]
\[-2 + \sqrt{6}\]
So, the given fraction is equal to \(-2 + \sqrt{6}\), which means \(m = -2\) and \(n = 1\).
The correct answer is:
B. -2, 1
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