Express \(\frac{8 - 3\sqrt{6}}{2\sqrt{3} + 3\sqrt{2}}\) in the form \(p\sqrt{3} + q\sqrt{2}\).
FURTHER MATHEMATICS
WAEC 2016
Express \(\frac{8 - 3\sqrt{6}}{2\sqrt{3} + 3\sqrt{2}}\) in the form \(p\sqrt{3} + q\sqrt{2}\).
- A. \(7\sqrt{3} - \frac{17\sqrt{2}}{3}\)
- B. \(7\sqrt{2} - \frac{17\sqrt{3}}{3}\)
- C. \(-7\sqrt{2} + \frac{17\sqrt{3}}{3}\)
- D. \(-7\sqrt{3} - \frac{17\sqrt{2}}{3}\)
Correct Answer: B. \(7\sqrt{2} - \frac{17\sqrt{3}}{3}\)
Explanation
Given \(\frac{8 - 3\sqrt{6}}{2\sqrt{3} + 3\sqrt{2}}\),
first, we rationalise by multiplying through with \(2\sqrt{3} - 3\sqrt{2}\) (the inverse of the denominator).
\((\frac{8 - 3\sqrt{6}}{2\sqrt{3} + 3\sqrt{2}})(\frac{2\sqrt{3} - 3\sqrt{2}}{2\sqrt{3} - 3\sqrt{2}})\)
= \(\frac{16\sqrt{3} - 24\sqrt{2} - 18\sqrt{2} + 18\sqrt{3}}{4(3) - 6\sqrt{6} + 6\sqrt{6} - 9(2)}\)
= \(\frac{34\sqrt{3} - 42\sqrt{2}}{-6} = 7\sqrt{2} - \frac{17\sqrt{3}}{3}\)
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