The expression 1-11+√3+11-√3 equals:

1956 AMC 12 Problem 11

Attempt Problem 11 of the 1956 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1956 AMC 12 solutions, or check the answer key.

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11.

The expression

1-\dfrac1{1+\sqrt3}+\dfrac1{1-\sqrt3}

equals:

1-\sqrt3

1

-\sqrt3

\sqrt3

1+\sqrt3

Combine the two fractional terms over

(1+\sqrt3)(1-\sqrt3)

The product of the conjugate denominators is

1-3=-2

Combining the fractional terms,

\begin{gathered} -\frac1{1+\sqrt3}+\frac1{1-\sqrt3} \\ =\frac{(1+\sqrt3)-(1-\sqrt3)} {(1+\sqrt3)(1-\sqrt3)}\\ =\frac{2\sqrt3}{-2}\\ =-\sqrt3. \end{gathered}

Adding the initial

1

gives

1-\sqrt3.

Thus, the correct answer is A.