Write this surd in its simplest form. 
Express $\dfrac{1}{2+ \sqrt3}$ in its simplest form.

NB: The textbook has the answer as $2 - \sqrt3$ but I can't see how that was achieved.
I tried $\dfrac{1}{2} + \dfrac{1}{\sqrt3}$ and multiplying the top and bottom by $\sqrt3 $ to get $\dfrac{1}{2} + \dfrac{\sqrt3}{3}$ so far.

Any time you are simplifying an expression like $$\frac{c}{a \pm \sqrt{b}},$$ multiply it with $$\frac{a\mp \sqrt{b}}{a\mp \sqrt{b}}$$ which gives you $$\frac{c(a\mp \sqrt{b})}{(a\pm \sqrt{b})(a\mp \sqrt{b})} = \frac{c(a\mp \sqrt{b})}{a^2 - b}$$ which has no more square roots in the denominator.
