$\left(\frac{2 + \sqrt{3}}{2 - \sqrt{3}} + \frac{2 - \sqrt{3}}{2 + \sqrt{3}} + \frac{\sqrt{3} - 1}{\sqrt{3} + 1}\right)$ simplifies to

### Concept & Formula This problem requires grouping terms smartly. The first two fractions are conjugates of each other. Summing them using a direct algebraic identity is much faster than rationalizing them individually. For conjugate fractions: $$ \frac{a + b}{a - b} + \frac{a - b}{a + b} = \frac{2(a^2 + b^2)}{a^2 - b^2} $$ ### Step-by-Step Solution Let's divide the expression into two parts to simplify calculation. **Part 1: The First Two Fractions** Let $E_1 = \frac{2 + \sqrt{3}}{2 - \sqrt{3}} + \frac{2 - \sqrt{3}}{2 + \sqrt{3}}$ Using the identity above where $a = 2$ and $b = \sqrt{3}$: $E_1 = \frac{2((2)^2 + (\sqrt{3})^2)}{(2)^2 - (\sqrt{3})^2}$ $E_1 = \frac{2(4 + 3)}{4 - 3}$ $E_1 = \frac{2(7)}{1} = 14$ **Part 2: The Third Fraction** Let $E_2 = \frac{\sqrt{3} - 1}{\sqrt{3} + 1}$ Rationalize the denominator by multiplying top and bottom by $(\sqrt{3} - 1)$: $E_2 = \frac{(\sqrt{3} - 1)^2}{(\sqrt{3} + 1)(\sqrt{3} - 1)}$ Expand the numerator and simplify the denominator: $E_2 = \frac{3 + 1 - 2\sqrt{3}}{3 - 1}$ $E_2 = \frac{4 - 2\sqrt{3}}{2}$ Divide both terms in the numerator by 2: $E_2 = 2 - \sqrt{3}$ **Total Expression:** Total $= E_1 + E_2 = 14 + (2 - \sqrt{3}) = 16 - \sqrt{3}$ ### Exam Strategy & Shortcut Memorize the structure $\frac{x + \sqrt{y}}{x - \sqrt{y}} + \frac{x - \sqrt{y}}{x + \sqrt{y}}$. It always results in an integer if $x$ and $y$ are integers. For the third term, knowing that $\frac{\sqrt{3} - 1}{\sqrt{3} + 1} = 2 - \sqrt{3}$ is a standard result (often used in Trigonometry for $\tan 15^\circ$) allows you to bypass the rationalization step entirely, turning this into a 15-second visual question. ### Common Pitfall Students often try to find a single massive common denominator for all three fractions at once. This leads to extremely heavy algebraic multiplication, making it highly probable to drop a sign or miscalculate a coefficient. Always solve conjugate pairs first. ### Final Answer **Therefore, the correct answer is $16 - \sqrt{3}$.**