What is the following quotient? the fraction with numerator 1 and denominator 1 plus square root of 3

Multiply the numerator and denominator of 1/(1 + √3) by the conjugate, 1 − √3. This gives (1 − √3)/(1 − 3) = (1 − √3)/(−2) = (√3 − 1)/2. That matches choice A, (−1 + √3)/2.

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