1956 AHSME Problems/Problem 11

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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 (ProblemsAnswer KeyResources)
Preceded by
Problem 10
Followed by
Problem 12
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