JAMB 1986 · UME · Q13✱✱

Simplify $\left(\frac{1}{\sqrt5 + \sqrt3} - \frac{1}{\sqrt5 - \sqrt3}\right) \times \frac{1}{\sqrt3}$ .

Worked solution (try it first)

Rationalise each fraction.
$\frac{1}{\sqrt5 + \sqrt3} = \frac{\sqrt5 - \sqrt3}{5 - 3}$ , which is $\frac{\sqrt5 - \sqrt3}{2}$ .
Likewise $\frac{1}{\sqrt5 - \sqrt3} = \frac{\sqrt5 + \sqrt3}{2}$ .
Subtract: $\dfrac{(\sqrt5 - \sqrt3) - (\sqrt5 + \sqrt3)}{2} = \dfrac{-2\sqrt3}{2}$ , which is $-\sqrt3$ .
Multiply by $\frac{1}{\sqrt3}$ : $-\sqrt3 \times \frac{1}{\sqrt3} = -1$ , option D.