# 1956 AHSME Problems/Problem 11

The expression $1 - \frac {1}{1 + \sqrt {3}} + \frac {1}{1 - \sqrt {3}}$ equals: $\textbf{(A)}\ 1-\sqrt{3}\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ -\sqrt{3}\qquad \textbf{(D)}\ \sqrt{3}\qquad \textbf{(E)}\ 1+\sqrt{3}$

## Solution

We can combine the second and third terms using the rules of arithmetic: We have $-\frac{1}{1 + \sqrt {3}} + \frac{1}{1 - \sqrt {3}} = \frac{-(1 - \sqrt{3}) + (1 + \sqrt{3})}{(1 - \sqrt{3})(1 + \sqrt{3})} = \frac{2\sqrt{3}}{1 - 3} = -\sqrt{3}$.

Therefore the entire expression is equal to $1 - \sqrt{3}$, so our answer is $\boxed{\textbf{A}}$ and we are done.

## See Also

 1956 AHSME (Problems • Answer Key • Resources) Preceded byProblem 10 Followed byProblem 12 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 All AHSME Problems and Solutions

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