JAMB Mathematics 2014 Paper

Concept:We rationalise the denominator by multiplying with its conjugate. This removes the surd from the denominator and lets us simplify the numerator.Explanation:Multiply the numerator and denominator by $\sqrt{2} - \sqrt{3}$.$$\frac{2\sqrt{2} - \sqrt{3}}{\sqrt{2} + \sqrt{3}} \times \frac{\sqrt{2} - \sqrt{3}}{\sqrt{2} - \sqrt{3}}$$In the denominator, use the difference of two squares:$$(\sqrt{2})^2 - (\sqrt{3})^2 = 2 - 3 = -1$$In the numerator, expand carefully:$$(2\sqrt{2})(\sqrt{2}) = 4$$$$(2\sqrt{2})(-\sqrt{3}) = -2\sqrt{6}$$$$(-\sqrt{3})(\sqrt{2}) = -\sqrt{6}$$$$(-\sqrt{3})(-\sqrt{3}) = 3$$So the numerator becomes:$$4 - 2\sqrt{6} - \sqrt{6} + 3 = 7 - 3\sqrt{6}$$Divide by $-1$:$$\frac{7 - 3\sqrt{6}}{-1} = -7 + 3\sqrt{6}$$Rearrange to match the options:$$3\sqrt{6} - 7$$Answer:A. $3\sqrt{6} - 7$