Find the value
\[ a = 2 + \sqrt{3} \]
Solution:
\[ \frac{1}{a} = \frac{1}{2 + \sqrt{3}} \times \frac{2 – \sqrt{3}}{2 – \sqrt{3}} \]
\[ = \frac{2 – \sqrt{3}}{4 – 3} = 2 – \sqrt{3} \]
\[ a – \frac{1}{a} = (2 + \sqrt{3}) – (2 – \sqrt{3}) \]
\[ = 2\sqrt{3} \]