The expression $1 - \frac{1}{1 + \sqrt{3}} + \frac{1}{1 - \sqrt{3}}$ equals

### Concept & Formula This problem requires combining fractions that have conjugate binomials in their denominators. Instead of rationalizing each fraction separately, it is much faster to find a common denominator for the fractional parts, leveraging the difference of squares formula: $$ (a + b)(a - b) = a^2 - b^2 $$ ### Step-by-Step Solution * **Calculation / Deduction:** * Let's isolate and simplify the fractional portion of the expression first to keep things clean. $$ - \frac{1}{1 + \sqrt{3}} + \frac{1}{1 - \sqrt{3}} $$ * The common denominator for these two fractions is $(1 + \sqrt{3})(1 - \sqrt{3})$. * Combine the numerators over this common denominator: $$ = \frac{-(1 - \sqrt{3}) + (1 + \sqrt{3})}{(1 + \sqrt{3})(1 - \sqrt{3})} $$ * Simplify the numerator by distributing the negative sign carefully: $$ = \frac{-1 + \sqrt{3} + 1 + \sqrt{3}}{1^2 - (\sqrt{3})^2} $$ $$ = \frac{2\sqrt{3}}{1 - 3} $$ $$ = \frac{2\sqrt{3}}{-2} $$ $$ = -\sqrt{3} $$ * Now, substitute this simplified result back into the original full expression: $$ 1 + (-\sqrt{3}) $$ $$ = 1 - \sqrt{3} $$ ### Exam Strategy & Shortcut Whenever you see a pair of fractions taking the form $\frac{x}{a+b} \pm \frac{y}{a-b}$, immediately cross-multiply to form a single fraction. The denominator instantly becomes a clean integer via difference of squares ($a^2 - b^2$), and the numerators will usually feature satisfying cancellations of integers or surds. Do not waste time rationalizing them as two separate, isolated problems. ### Common Pitfall Many students lose track of the leading negative sign on the first fraction: $- \frac{1}{1 + \sqrt{3}}$. When finding the common denominator, they might forget to distribute that negative to *both* terms in $(1 - \sqrt{3})$, mistakenly writing $-1 - \sqrt{3}$ instead of $-1 + \sqrt{3}$ in the combined numerator. Always use parentheses when carrying a negative sign across a combined fraction. ### Final Answer **Therefore, the correct answer is $1 - \sqrt{3}$.**